Subverted in Dr. Horrible's Sing-Along Blog. The title character is painfully shy, and has been quietly in love with a girl from his laundromat for months. Ironically, his alter ego is a megalomaniacal villain who wants to rule the world.
Nessiah in Dept Heaven Apocrypha almost never talks to people he doesn't like and trust—in addition to being very shy, he's been through excessiveamountsof traumarecently, and is still recovering from aphasia, which makes it difficult for him to string full sentences together anyway. He's still his usual dere self around Milanor, though.
Christine on The War Comms, big time. Her sentences more often than not end with "um...please don't hurt me for saying so...".
Gloria Benson in V4 of Survival of the Fittest. She's always whispering, stuttering, and blushing when she speaks. Even when she talks to her friends, she's painfully shy/quiet, and keeps apologizing for what she says. Lucy Ashmore, also from v4, qualifies as well. She's just as shy, due to bullying. Sometimes this slides into Cute Clumsy Girl territory as well, due to her nervousness.
This youtube account has documentaries on what it's like to have the DSM recognized personality disorders including the avoidant.
The Autobiography of Jane Eyre: Jane is a quiet and lonely girl who doesn't have any friends and any good family. She's nice, but has No Social Skills. It's quite unusual that such a girl would start vlogging and sharing her life with the internet. She considers it a challenge and a brave thing to do, and she would like to get some help from her viewers.
We Are Our Avatars: At first, Marcia did not like talking with the others. But she warms up to them and becomes more friendly over time.
Jen Delaney of The Jenkinsverse was a nobody with a dead-end job in I.T. whose entire love life consisted of one ill-advised drunken fumble at the office Christmas party... prior to her Alien Abduction. Getting away from Earth seems to have agreed with her, transforming her into a confident, competent, self-described "Space-Babe".
The Porcelain Doctor from Dead West plays with this. His aura, The Devil's Veil compells people to think he is this, and a Fragile Flower to boot. Sure he is reserved and quiet, but he is a military surgeon, leader of the medical corps, and if on the field, he is not shy about shouting and ordering people around. He is also not the kind whom you can easily push around.
Lauchlan, the main character of Mix Beer With Liquor And You Will Get Sicker is painfully shy, socially inept and introverted to a fault. He has no friends beyond his immediate family, his neighbor's housemaid, his cat and the company horses. His self-esteem issues and crushing lack of social skills makes something as simple as talking to shopkeepers rather harrowing. Add in that he's a Nervous Wreck, and Word of God declaring that he's on the Autism Spectrum, and he's practically got a whole bouquet.
Miss Martian is this way in DC Superhero Girls. When introduced to Wonder Woman in episode 1 she turns invisible and she later does the same when Poison Ivy tries to talk to her. Apparently this is common for her and the only person she seems fine around is Starfire.
Fey: A downplayed example, and a trope she develops out of. She speaks softly initially when she first arrives at Whateley Academy, as said in Welcome to Poe:
Belle nodded then turned to Fey. "Your turn, hon." Fey drew in a breath, then started out softly. [...] "Could you speak just a bit louder, dear?" Belle questioned with an encouraging smile. "You have a beautiful voice and shouldn't be ashamed of it at all. Plus, your grammar is pretty good so far, by the way.
Tennyo: She's is actually quite shy until she gets to know you, which other people see as aloof and feeds into her "All of the Other Reindeer" problems.
Lapin: Currently suffering from PTSD, depression, and a generally poor self image, which all conspire to make her less than assertive, especially under the influence of her nymphomania meds, but Dr. Bellows judges that it's best to control the nymphomania and try to rebuild her self esteem using more conventional therapy.