Despite having lost an eye, an arm, and a leg, Skeleton Jorge is not slowed down by these infirmities, nor does it diminish how awesome a swordsman he is.
Joaquin's eye gets damaged in the climax, resulting in him wearing an eyepatch.
In Disney/Pixar's Brave, Merida's father, King Fergus, loses a leg in a fight against a bear and has it replaced with a wooden peg leg. He also punched said bear in the face later.
Finding Nemo had a few of these, including the title character with his underdeveloped fin. There's also Dory who suffers short-term memory loss and Gill, the leader of the tank gang who, like Nemo, has a permanently injured fin. If you count Marlin's crippling fear of the ocean (not undeserved), then he qualifies as well.
Gobber from How to Train Your Dragon only has one leg and one arm. As a Viking, he is very badass. Toothless only has half a tail but is still badass and at the end, Hiccup becomes one when his lower left leg is destroyed during the final battle. Drago Bludvist from the sequel counts too, having lost an arm after being attacked by a dragon. Although he's badass in a verydifferentway to the characters above.
In Kubo and the Two Strings, most of the main cast counts — Kubo has only one eye, while his mother suffers from complications due to head trauma which leaves her comatose during the day. Beetle has amnesia, and has difficulty remembering even basic things without Kubo and Monkey to keep him focused. Monkey is injured midway through the movie and is Secretly Dying, as the magic that keeps her soul in her body fades. The Moon King is completely blind until the ending, where Kubo's other eye gives him sight at the cost of his memories.
Viper in the Kung Fu Panda franchise was, according to Secrets of the Furious Five, born without fangs and learned kung fu to make up for it. She also lacks limbs, but this doesn't really count as a handicap, what with her being a snake.
In Mulan, Mulan's father Fa Zhou demonstrates perfect sword form and his mind is still sharp even in his advanced age. Unfortunately, his age, bad leg, and implied heart problems make him incapable of fighting for more than about a minute, which was why Mulan chose to take his place in the army.
From Moana, we have Tamatoa. A demigod ripped off one of his legs. But this crab can and does humiliate said demigod later. It helps that the demigod was rusty.