Need Shrinking Violet personified? Well, look no further than the appropriately-named Violet Parr from The Incredibles. Not only is she incredibly shy, but she also has the hair, the mumble, and a desire to be just another girl, and she can't even look the school heartthrob, Tony Rydiger, in the eye. Her family are superheroes in hiding, and her powers are invisibility and forcefields—Word of God is that these are symbolic of her personality, respectively being her wish to hide from everyone else and her need for insulation from the rest of the world. Needless to say, she embraces her abnormalities, gets an Adrenaline Makeover, and puts her hair back in the process of helping her family save the world.
And to top it all off, by the end of the movie, it's Tony who can't look Violet in the eye, instead of the reverse. Think about it.
She also discovers the ability to levitate in the center of her force field when forced to save her brother. Both seem equally surprised.
A rare male, yet non-Butt Monkey example: Austin from The Backyardigans also embodies this trope. Not only he's purple (violet), during season 1 he was very shy and didn't speak a lot (he had songs about that). Over the course of the series, he becomes more outgoing and gets dynamic lead roles.
Stacy from Daria, the shy and sweetest member of Quinn's Fashion Club.
Male example: Shifty Dingo in the 2nd season of Blinky Bill after his Heel-Face Turn easily qualifies as one of these. He's much more quiet than the rest of the gang and also has a Lovable Coward side to him.
Another male (kinda) example from Futurama would be Kif Kroker, who is usually a Deadpan Snarker when dealing with his boss ZappBrannigan, but seems to show this side when his Love Interest Amy is around. In the episode that brought them together as a couple, she kept getting phone calls of someone breathing and panting heavily on the other end; she assumed they were prank calls, when it was really Kif calling to tell her how he feels, but too scared to say anything, thus hyperventilating until he hung up.
The Funny Company, a syndicated series from the early 1960s, featured a girl named Violet who really did shrink down to doll size when she felt self-conscious.
In the Disney adaption of Alice in Wonderland, when Alice stumbles on to a talking garden, they promised her that they would sing a song to her. All the flowers wanted to sing about themselves. The violets wanted to sing about themselves, and when they brought up the suggestion, they were quiet and they slowly backed up in the shade below the other flowers.
Coco Pommel, the Beleaguered Assistant to the Jerkass-ish Suri Polomare, is incredibly submissive to her boss as she's forced to do all her dirty work and get her coffee. At the end of the episode she appears in, Coco musters up the will to quit her job.
Cheese Sandwich, the only pony who can give Pinkie Pie a run for her money in terms of Fun Ponified, and the pony equivalent of "Weird Al" Yankovic, admits to being painfully shy when he was a colt. It was around that time that he met Pinkie and was inspired to make a name for himself as the craziest party pony in Equestria. Even then, he still travels Equestria with only his rubber chicken as a companion.
Doug Funnie, the title character of Doug, is a more subtle male example. However, it usually depends on the episode.
Schoolhouse Rock has an example in Mr. Morton (the subject of the sentence; and what the predicate says he does) who is played entirely sympathetically. His entire song is about how he's trying to get his neighbor, Pearl, to like him but is way too scared to ask her out or even talk to her, including two lines that explicitly state "Mr. Morton was very shy" and "Mr. Morton was very nervous". Eventually, she ends up proposing to him, and they live Happily Ever After.
Filburt from Rocko's Modern Life. He's even a turtle (as well as being named after a nut) with a literal shell to go along with it, which he will often retreat into in times of social stress. In one episode, he manages to get past this and become a lounge singer, although not without first feeling incredibly self-conscious even that Rocko heard him singing in his own trailer, and suffering from a very intense amount of stage fright. He does eventually get a Love Interest, though when he plans to propose to her she succeeds in proposing to him first.
Princess Aurora/Briar Rose of Sleeping Beauty stutters and can hardly get out a word when a stranger stumbles across her singing.