YES! Definitely!
It’s true that I’m a bit of an extrovert, but I, too, have anxiety, and I know many TV writers who are introverts or have anxiety. It is possible! In fact, I’d say writing is a career well suited for you, haha. You’ll have good company.
The best part about being a TV writer is you have an agent, and they’re the person who’ll negotiate all your deals for you, and talk you up to other people on the phone (oh god, the phone!) so you don’t have to. They’ll schmooze the town for you and send out your script, and tell everyone you’re great, while you’re at home, in your pj’s, working on your next great pilot. It’s the best.
The tricky part for socially anxious folks is getting that agent in the first place. You have have the best scripts in the world, but if you don’t meet people and send them out, it’ll be hard for anyone to know about you in order to give you a job. So you have to do some amount of networking/socializing. The good news is, networking can look like lots of different things! It doesn’t have to be going to one of those horrible networking events where everyone’s handing out business cards (dear god, never). It can mean hanging out with your friends, talking about your lives and your projects. It can mean joining a book club or a D&D group, or especially a writing group.
You can also submit your scripts to contests and fellowships (particularly Nicholl, the various TV Fellowships, Austin Film Festival, and the Black List).
Social media is your friend, you can meet people in private Facebook groups (srsly like 90% of my Facebooking is done in private groups now), or on Twitter, or on Tumblr. Some of my good friends are from Tumblr, and from fandom. I wrote a letter of recommendation for film school for a Tumblr friend. I’ve recommended Tumblr friends for jobs. I’ve donated to Tumblr friends’ kickstarters, read their scripts, mourned for the loss of their pets… all sometimes without meeting in person.
Once you have an agent, come back and we can talk about strategies for socially anxious people taking meetings (they’re like job interviews but waaaay more casual – you get to wear jeans and sneakers, and poop jokes are fair game.) and other tricky things to get through as an anxious person.
It’s possible! And worth it!