I've eaten there many times because it's a family favorite, and I've never had any problems getting vegetarian food. Basically, you can call to the Hoop Dee Doo Revue ahead of time, or probably even let them know when you check in, and definitely talk to your server as well. If you can't figure out how to call the Hoop Dee Doo yourself, I'd recommend going to guest services in whatever park you're in the day of your reservation first thing in the morning and telling them your situation and asking what needs to happen to request a vegetarian meal for that night.
They've always given me a pretty boring plate of pasta with marinara sauce and some mix of vegetables. Not my favorite food ever, but it was edible, and vegetarian. I also never tried requesting a specific food, just something vegetarian, but according to what I've read here you can probably request anything that is on the buffet at Trails End next door.