Hi Amanda,
This is a great question. As someone who contacts Disney Dining for my special diet allergies, I can attest to the vast menu options available for any diet concerns. For awesome vegetarian meals, I would suggest
Teppan Edo in Epcot and
'Ohana at Disney's Polynesian Village Resort. Both are uniquely themed dining locations with delicious offerings. Teppan Edo offers a teppan-yaki style meal where the skilled chef will perform (and cook!) for tables of eight. 'Ohana offers Hawaiian delicacies and will typically prepare scrumptious platters of tempura veggies, rice and tofu upon request. The live entertainment and family atmosphere at 'Ohana make this a "must-do" for us! You can't go wrong booking buffet style meals like The Crystal Palace or Chef Mickey's, where vegetarian options are always plentiful.
If you don't see a vegetarian option at counter service locations (or any menu, for that matter), simply ask! The Cast Members are well versed in accommodating vegetarian meals and will come up with alternatives.
I'm limited in space, Amanda, but I have the utmost confidence Walt Disney World dining will not disappoint. Let me know if you need further suggestions and I'll be happy to help.
Sherri