I currently work at a church nursery, and I haven't had a problem with my anxiety while working yet. I only have to work a few days a week for 2-3 hours a day, so its not stressful. I only have to interact with a few people at a time as well, and I didn't have to have an interview, just apply and talk to the preacher. If you do well with children, this might be a good job for you.
Since you mentioned that you like animals, you could look into helping at a vet office as an assistant. As far as I know, you don't need medical training to work at the front desk. Keep in mind, the smaller the lobby, the less people it can hold at a time, the less chance you have at getting anxious.