Sokka is the only non-bender of the main group. Despite his near-uselessness in battle and lack of spirituality (and role as the comic relief), as the show progresses, we see his engineering and strategy prowess improve to the point where he is openly labeled as the "idea guy" of the team. More recently, his inferiority in combat spurred him to seek out training in that area, plus he acquired a unique sword, making him even more badass despite the fact that he is still inherently normal.
Also, he took down Combustion Man, a Determinator so mean that the Bender part of the team could barely slow him down.
Of the Order of the White Lotus members who we see fight, only one is a non-bender, Piandao, who, not so incidentally, is the one who gave Sokka his training, and more than holds his own. To reiterate: a foot soldier in the Fire Nation achieved the rank of swordsmaster, then retired—and successfully resisted all attempts to unretire him. A force of 100 soldiers was sent after him. He sent them all back.
When you think about it, any non-bender who wants to be taken seriously in this world needs to be this trope, because in a world where people can shoot fire out of their freaking hands, it must take a lot to get noticed. Considering how tiny the Southern Water Tribe is, and that they lack waterbenders, it's quite something that Hakoda's group survived for over two years to mount an attack with Team Avatar on the Day of Black Sun. The Chief seems to be a very smart and respected Badass Normal himself.
There's even Badass Normalmooks in the form of the Yu Yan Archers. These guys actually managed to capture Aang by themselves, with nothing but their own impressive archery skills.
Speaking of archers, Longshot is pretty Badass Normal. Actually, all of the Freedom Fighters can be considered this.
Especially Jet. Yes, he's a well-intentioned extremist, but he's incredibly skilled with his Twin Tiger hookswords, to the point where he can defeat an (admittedly in-training) Avatar. If Katara hadn't gotten involved, Jet would have won. He's smart, he's cocky, he can hold his own against powerful benders, and he gives us one of the best fight scenes in the entire show (a personal favorite of Mike and Bryan's) when he has a hooksword vs. double broadsword battle with an incognito Zuko. We still don't know who would have won if the Dai Li hadn't interfered.
Also the pirates from season one, who manage fairly easily capture Aang and Sokka, and later manage to hold their own with Zuko and his fellow fire bender with just smoke bombs and normal (if some what exotic) close range weapons.
From The Legend of Korra, we've got an organization, the Equalists, who are trying to eradicate bending completely. They are led by the Big Bad Amon, a Magnificent Bastard through and through, who has no problem getting his own hands dirty. He gives numerous benders a chance to fight back before he takes away their bending. None of them succeed in fighting him off. Not even Korra; luckily, he could not take the element she did not yet have. However, this is subverted in the season finale, where it is revealed that Amon (real name Noatak) is actually a waterbender and more importantly, a bloodbender. When he accidentally reveals this to all of Republic City, he and his revolution loses all credibility and he is forced to flee the city.
Amon has frighteningly competent henchmen. The Equalists have no lack of Elite Mooks, and they use clever strategies and strength in numbers to overwhelm their bending opponents. Even without those, just a few of them can put up a fight against powerful benders. They also have new technology on their side, most distinctively the electroshock weapons that put opponents out of commission quickly, and are particularly effective against the metal-wielding, metal-armored police force.
Asami Sato serves as the group's resident Badass Normal, being a talented martial artist who's quite handy with an Equalist Power Fist.
Zaheer, the Big Bad of Season 3 is the leader of a group of benders dedicated to killing the Avatar and is treated as equally dangerous as his crew despite the fact that prior to Harmonic Convergence he was a non-bender. However, after the events of Season 2, he gains the ability to Airbend.
In season four, Varrick even gets a few moments. Though he's not skilled in combat, he manages to take out multiple mecha tanks on his own just by rigging what looks like a generator.
Bumi, the youngest of Aang's children, spent most of his life as one as he was the only one of his siblings unable to bend. He actually developed a complex because he was a non-bender born to an Avatar, but he was determined to prove himself in other ways. You don't become a commander of the United Forces without being a badass. Like Zaheer, he gains Airbending late in his life, as well.
Casey Jones from Teenage Mutant Ninja Turtles fits this trope. He manages to take down street gangs, alien triceratops and robots with just a hockey stick, two baseball bats and a golf club, while all he has for protection is a hockey mask.
Get Ed has Ol' Skool, the old Mentor of the main group of heroes. Every other character has high tech Hoverboards, Jetboots, Jetpacks, etc. and top-of-the-line protective gear to use. Ol' Skool has a conventional skateboard & a simple construction helmet. With just these things he is able to outrace anything in the series without getting hurt or breaking a sweat.
In Winx Club, whether or not the Red Fountain Specialists fit under this trope seems to depend on the plot. Sometimes they're smacked around by wild monsters, and sometimes they're holding their own against major threats like the Trix and the Wizards of the Black Circle.
The Winx themselves are this in the spinoff World of Winx where, because they must go undercover, they must deal with any threats, using their fairy powers only as a last resort.
W.I.T.C.H.: Caleb never has any more powers than your basic fit young man, yet he somehow is just as effective in a fight as the main girls who, kinda, control the essence of the five elements! By the end of the second series, even Will's pet dormouse has more powers than him, and he still manages to hold his own in a fight. If that isn't badass, nothing is.
Caleb is sometimes more effective than all of the main girls put together.
In season 1, Will more or less is a Badass Normal, as her powers hold no use in battle. So the writers have to improvise, giving her what nonmagical actions she needs to fight the bad guys. However, in the second season, once she gains the power of Quintessence, she no longer fits this trope.
Robin from Teen Titans, though it frequently escalates into the Charles Atlas Superpower level. Slade is a villainous Badass Normal, except in the fourth season where an intergalactic demon resurrects him and grants him fire powers, removing the "normal" - until his powers are removed again, and he returns to Badass Normal status by killing a flaming-axe-wielding guardian of the underworld with nothing but tactics and martial arts.
There's also Red X, who utterly curbstomps all of the Titans at once with barely any effort at all, and that was the first time he faced them. True, he has the stolen suit that Robin created, but the Titans have formidable powers and Robin has all of his own gadgets, and they still could barely touch him.
Hobgoblin in Spider-Man: The Animated Series. In this universe, he predates the Green Goblin, and while his weapons came from Norman Osborn there was no strength-enhancing formula involved. Unfortunately for him, when the Green Goblin does turn up, he's at least as strong as the comics version.
In the Justice League, we have Batman and The Joker as a heroic and villainous example respectively both of whom have superpowered allies and enemies alike intimidated by and afraid of them. For good reason, too: they both regularly take down said superpowered individuals with nothing more than cunning, intelligence, and some gadgets. So much so that in one special when Joker shifts his attention from Batman to Superman, he refers to it as "playing on easy mode for a change."
Justice League Unlimited made sure the normals were all badass in their own way. Though it was a sore spot for Wildcat, believing that just with no superpowers besides being an excellent fighter made him obsolete in the League. He even said that while Black Canary Fights Like a Normal, the fact she had the canary cry sonic scream made her much more useful to the team.
The Question, who is often nearly as many mental steps ahead of everyone as Batman. He's a consistently paranoid conspiracy theorist, but he lives in a world where conspiracies are a lot more likely.
All his theories that were actually put to the test on-screen turned out to be right. He's 3-for-3 so far, who's to say he's wrong about the rest?
He's also right that Baskin Robbins had a secret 32nd flavor. The significance of this isn't quite clear.
In the episode "Tabula Rasa" of Justice League, the Justice League battle Amazo, an android with the ability to copy any superpower. He defeats the Justice League one by one. Possessing Superman's powers, he turns to Batman to scan him:
Amazo: You don't have any special powers. Batman: I have this. (pulls out kryptonite) It's a package deal. You get our strengths; you get our weaknesses.
In the episode "Patriot Act", a tribute to the Seven Soldiers of Victory, and Badass Normals in general.
"Panic in the Sky", too, where not only do three of the League's badass normals take out at least five clones, including a super-sized Apache Chief wannabe, but the Watchtower staff fights a T. rex. The jury's out on whether or not they survived that, but the mere fact that they tried is badass incarnate.
"Taskforce X" uses this trope to help make the Villain Protagonists of the episode more sympathetic. Even if they are evil, watching four normal people (with a fifth as mission control) infiltrate the Watchtower and make off with a giant magical suit of armour is still sort of cool.
Speaking of which: Rick Flag, asskicker extraordinaire. Of the four members, three carry weapons. Flag does not. When questioned about this, the simple explanation is "he doesn't need a weapon." He more than lives up to that hype.
The head of Cadmus Amanda Waller. Not only is she able to intimidate Batman, she has absolutely no combat training, and yet still goes up against Brainthor along with the original seven with nothing but a handgun, even pulling a brief Back-to-Back Badasses with Green Lantern.
Green Lantern gets this himself in 'A Savage Time'. The League is sent back to WWII, where GL's ring runs out and he meets up with a group of Allied soldiers. After proving himself by beating down TheBigGuy, he identifies that he was a member of the U.S. Marine Corps. He spends the rest of the movie kicking ass with no super powers what-so-ever.
Darel, the protagonist from Kulipari: An Army of Frogs, is not a Kulipari himself. Nevertheless, he's a brave warrior who constantly trains to be like a Kulipari, and he always manages to hold his own during intense fights.
Band manager Charles Foster Ofdensen of Metalocalypse, who can kick the ass of a man twice his size and be thrown from a top story window and land like a cat. He also runs the seventh largest financial empire in the world.
Transformers Animated: Captain Fanzone gets an honorable mention here for managing to hold off one of Meltdown's mutated monstrosities by himself (while the other one gave the robot ninja trouble). Despite his trouble with technology and usual Butt Monkey status, he's still a brave and competent police chief who doesn't let his early prejudice against the Autobots get in the way of upholding the law.
And on the organophobic Cybertron he's perceived as a walking bio-terrorism weapon, something he puts to good use to interrogate a robot four times his size.
Sari. Sure she's got her all spark key, but there's been several times when it doesn't work and she finds a way to beat the bad guys anyway. Point in case: survival of the fittest - uses the moves Prowl taught her to distract meltdown, Nature calls - manages to escape from and then save zombified Prowl and Bumblebee, Megatron Rising - tricks Blackarachnia into walking in circles, Sari no ones home - fights off the constructicons (admittedly she did use her key several times here, but moments like rollerblading through those automaton arms at full speed still count!)
Mighty Max and his allies Virgil and Norman all count, as they often had to defeat super-powered villains while possessing no actual powers of their own aside from Norman's great strength and swordfighting skills.
Although Norman is a LOT older than he looks and is immortal (but not invulnerable), so whether or not he qualifies as normal is debatable.
Jackie from Jackie Chan Adventures. He pulls off many stunts including the classic Improvised Weapons and a great deal of other things that would be impossible if he weren't a cartoon character. He also doesn't use the magic on the show as much as other characters, using his own skills instead. Lampshaded by Ratso when Chan ran back up a train as it was falling off a cliff.
"No way he's human."
Jade as well, in an Offscreen Moment of Awesome Jade was able to beat up a black belt that was at least 4 times her height and musclemass.
She also is often able to solve complex problems and schemes that are able to defeat superpowered beings.
Uncle, Tohru, Viper, El Toro, Valmont and Chi Fu also have moments of badass normalcy.
Captain Black. He gets bonus points for being the show's resident skeptic.
Jake's Dad in American Dragon: Jake Long is this. Jake is a dragon, Rose is a born dragon hunter Action Girl, Spudis the embodiment of a Genius Ditz, and Trixie just has street smarts. Jake's Dad is a Cougar Scout, has extreme parent insticts, and that's it. However, he unknowingly defeated the Jersey Devil with a trombone and a can of bear spray. He also held his own against badass dragon blood drinking vampires for over four minutes with his bare hands in the episode, "Bite Father, Bite Son". During the final episode, he used a vase to kill an army of demons. Pretty good, considering he did it without knowing his son was a dragon (except the last one and he had just learned). All the above had powers, he didn't.
Arguably, the Huntsclan considered themselves normals.
Elisa Maza in Gargoyles. She can't fly, break rocks with her fingers, zap things, or even have access to high technology like practically everybody else, but all she needs is her handgun, a running start, and an attitude to help take down characters with far more advantages than she has. Sure, she loses quite often, but then so does everybody.
Probably the best example is King Arthur Pendragon, who manages to defeat Macbeth (who is immortal and virtually unkillable) with his powered suit in outright combat one on one, something Goliath was never able to do despite his physical advantage. He did this without Excalibur, just an ordinary mace.
In WildC.A.T.s Grifter is the only member of the team who is full-blooded human, while the rest have superpowers due to being descendants or full-blooded members of an ancient alien super-race. His only powers are trickiness, good aim, and general badassery.
In the original comic, Grifter was a member of Team 7, and all of Team 7 were experimental super soldiers with super abilities. The powers seemed to not be used much later, except for Backlash.
Quackerjack manages to be a "supervillain" without actual superpowers. When NegaDuck steals the powers of a number of other supervillains, which includes Quackerjack, he gets his "wackiness".
One episode made fun of this trope. It featured Darkwing going to a planet where everybody had superpowers except for one normal guy. As he is one himself, he is told to take on the role of the normal guy. Normal Guy eventually shows up and builds a machine to steal everyone else's superpowers.
Silver Sable and Black Cat are both examples of this from The Spectacular Spider-Man. Tombstone is possibly an example- he is never explicitly stated to have superstrength, but he is shown to be able to trounce Spider-man effortlessly when taking the latter by surprise, and be a match for him even a straight fight. Hammerhead is a partial example- he mostly relies on his strength and skill at brawling, but uses his steel-plated skull to absorb any punishment a normal man couldn't take. The Enforcers also qualify in their first appearance, though later they get upgraded with Powered Armor. Kraven was one, but has now acquired superpowers.
Red Witch's fanfic "Tangled Web" uses this to great effect. In the fight against the Black Rose, The Dragon factored in bionics, psionics, and biodefenses...only to have the hacker hand him his ass.
There's also Ambadassador Zozo. He's from a race of Hobbits, stands about a meter high, very cheerful, looks utterly harmless...but he's wicked fast, a great shot, and very sneaky.
Danny Phantom. In a World... where ghosts roam every corner, nearly all the ghost hunters shown are incompetent... except Valerie Gray who stands out among them as a normal gal packing (anti-ghost) heat (and being a ninth-degree black belt). Sam Manson gets a few of these moments throughout the series as well, but not Tucker. He's more of the Action Survivor.
Danny's mom probably qualifies. And she does it without breaking her matronly demeanor! Jazz has also proven curiously capable at piloting her dad's powered armor, so it must run in the estrogen.
Even Jack has his moments. He defeated Plasmius to save his family in "Million Dollar Ghost" and saved Danny from Skulker in "Girls' Night Out".
Terry McGinnis in Batman Beyond. Yes, he has the suit. But he has taken on explicitly superpowered villains without it, including psychics, shapeshifters, aliens, and mix-and-match creatures, even when he is at a distinct disadvantage. And won. He, armed only with one of his mentor's utility belts, also went up against his own suit when it was taken over by a rogue AI. He won that, too.
It's been stated in series that Terry is a better Batman than Bruce.
Special Agent Fowler from Transformers Prime, who can last a few minutes on Soundwave in a fight with just a helicopter, and who survived being tortured repeatedly with an energon-powered transformer-sized cattle prod and still had enough badass in him to mouth off to his torturer.
Jack Darby. Regular teenager, works at a burger joint...oh, and has successfully defeated a psychopath Decepticon with nothing but some energon and a multi-tool, traveled to Cybertron, and various other acts of surprising badassery.
And was, temporarily, entrusted with the Matrix of Leadership, which arguably made him the first human Prime.
Lt. Dwayne Hunter in Big Guy and Rusty the Boy Robot proves his military skills in Double-Time part one, rigging up a pair of Arm Cannons powered by an Iron Man style chest piece in an abandoned mechanics shop to take out four Seek and Destroy scouts from Legion Ex Machina after the Big Guy suit has been heavily damaged.
Xiaolin Showdown brings us self proclaimed boy genius Jack Spicer, who is generally too incompetent to be holding a lot of the show's MacGuffins, too lazy to learn physics bending Xiaolin arts or Heylin sorcery, and about as fit as you'd expect from someone who travels exclusively by helicopter backpack. Now -WHO- actually ends up taking over the world in the alternate future?. It's important to note, though, that he does show great talent with robotic and programming, and was able to construct some impressive machines to replace magical artifacts. He's also surprisingly skilled in winter sports.
Megan was just a normal preteen girl, surrounded by magical talking horses and a variety of fantasy creatures... but she fought a giant dragon with her bare hands, and killed Tirek, the most evil and violent villain in the show.
Sunset Shimmer. The typical serious threat to Equestria is a powerful immortal sorcerer, a possessed demigoddess, the sovereign of a militaristic society, or several of these at once. Sunset Shimmer was just Princess Celestia's former personal student, but her cunning and good preparation allowed her to come closer to conquering Equestria than some of its conventional enemies did. (But she never quite envisioned conquering Equestria, and turned herself around after seeing what her ambition had driven her to; since then, she's been on the Princess' side, and has accomplished quite a lot on Equestria's behalf.)
Earth ponies — normal (if pastel) ponies, as opposed to unicorns or pegasi — looked like this before it was confirmed that they have magically-enhanced strength, stamina, and skill with animals and plants. But they can't fly and they don't have telekinesis, so they pretty much still count.
Six. In a world of Nanite-created EVO monsters he's just an ordinary human, but with his ninja skills and Magna-Blades he's able to hold his own against them all, even briefly fending off the Big Bad, Van Kleiss, along with his Pack.
This as well as Dos, Trey, IV and Five. They aren't known as the 5 most dangerous people on the planet for a reason.
The KND are a paramilitary organizating entirely of kids, who fight villians who either have superpowers, or use technology that's to be considered a lot more advanced (or at least more sophisticated) when compared to the KND's 2x4 technology. While the KND, on the other hand, use strictly military training and the aforementioned 2x4 tech when engaging in combat.
Definitely Lucas Amato from Cybersix counts. He saves the life of the titular heroine almost as many times as she saves his, despite having no super powers to speak of.
Also Yashimoto, Meridiana's most renowned detective from the episode "Yashimoto, Private Eye," who effortlessly figures out Cybersix's secret identity after being blackmailed into doing so.
A villainous example - Belphegor, from the animated series of the same name. Aside from using a Ring of Power as his only weapon (and the occasional smoke bomb for escape purposes), he has no super powers to speak of. And he uses said ring only to tranquilize people. How does he always get away without guns? Well, he's strong, smart and always has an escape plan or a Batman Gambit ready. Many times he doesn't even make use of his underlings' help, but handles the most dangerous situations himself.
Cerina from Skysurfer Strike Force, who may be a cyborg, but unlike the rest of the Bioborgs, has no noticeable special powers.
Finn. He is a young teenaged human boy, however he fights off against evil demons and performs amazing athletic feats almost every episode. The entire series is effectively about him fitting into this trope.
Princess Bubblegum, officially as of "Lady & Peebles" when she beats the muscular Ricardio to a pulp and carries 4 people, including Lady Rainicorn who is a 15-foot-long unicorn, across the Ice Kingdom to the Candy Kingdom over 2 days. Also shows Badass traits in "What Have You Done?" where she is willing to beat screams out of the Ice King to cure the candy people of infection. Was once a close friend of Marceline, the Vampire Queen, and is even relatively confident enough to openly insult her, while Finn and Jake are quietly terrified of Marceline (although they are also her friends). Somehow managed to convince the evil Flame King to lock up his own daughter.
Warrior Princess and Clarence, who once led armies against each other in the past and battled to the death, with Clarence killing Warrior Princess who became Ghost Princess.
Joshua, Jake's dad, as it is implied he was an adventuring demon slayer in the past, and he has no powers, unlike Jake.
Joshua makes up for his lack of being a magical dog by being a dog that knows magic.
In Skyland there's Mahad and Dahlia who, despite not having powers like Mahad's sister Lena, can certainly hold their own against a Sphere patrol. Cortez, although primarily involved in being the leader of the rebel base of Puerto Angel, is no slouch either.
In Wolverine and the X-Men, there is an episode where a team of non-mutant ninjas attack Xavier's mansion and manage to neutralize all the X-men without being noticed. Those guys from Mutant Response Divisions should have hired ninjas to hunt mutants instead of building giant semi-sentient robots.
Princess Sally in Sonic SatAM doesn't have Sonic's Super Speed or Bunnie's Super Strength, much less any superpowers at all. But she is the leader of the Freedom Fighters who is a skilled strategist, excellent hacker, and of course, a competent fighter (although she has her moments where she ends up a Damsel in Distress).
Spoofed in a Tiny Toon Adventures episode parodying DC comics. Batduck (Plucky Duck) is being inducted into the Just-Us League of Supertoons, whose members mock him for having no superpowers other than his immense ego. However, his lack of superpowers helps him defeat supervillain Wex Wuther (Montana Max) who steals the superpowers of everyone else. For this feat, Batduck becomes the valet for the League members.
Gravity Falls, the Pines family face dozens of supernatural creatures, and beat them with only their wits, and what ever they can get their hands on.
Special mention goes to Soos' new Love Interest Melody, who started beating up Giffany-controlled animatromicson her first enocunter with the supernatural.
Young Justice has many, but a special mention has to go out to Artemis and Robin I/Dick/Nightwing. They turn fight scenes into an art form. When they were fighting together in Homefront, they worked seamlessly together against androids that had taken out their superpowered friends.
In Star vs. the Forces of Evil, Marco Diaz has good enough karate skills to defeat Ludo's minions. Star Butterfly herself used to fight with her bare hands before she gained her magic wand.
The SWAT Kats are two ex-police officers who, despite being Reassigned to Antarctica, manage to construct their own Cool Plane, the TurboKat, from junkyard scrap, alongside a bunch of other vehicles and gear, and manage to save the day. Deputy Mayor Callie Briggs, Enforcer Commander Ulysses Feral, and his niece, Lt. Felina Feral, are also quite capable of standing up to the villainry (though Commander Feral is against the SWAT Kats because of their vigilante activities).