I second the above posts. There are so many dining posts I could link here but just type "vegetarian at Disney" on a search engine and you'll see BOATLOADS of dining reports.
To give you some examples, though:
Most resort foods courts: cheese pizzas/flatbreads, sushi, salads, lo mein, lasagna, breadsticks, veggie burgers and fries, and, of course, most desserts and sides (with the exception of the rice krispy treats and zebra domes, etc. which have gelatin)
Most CS: veggie burgers/wraps/sandwiches/subs
You'll have the best luck at the TS restaurants and buffets for finding diverse meals. Even the meatiest of Disney restaurants, Le Cellier and Yachtsman, have great vegetarian options.
Some of my favorite choices at TS:
-50's Primetime: spaghetti and nutballs
-Sanaa: vegetarian sampler and samosas
-Boma: African/Indian buffet
and Crystal Palace: Mediterranean-based buffet
and Tusker House: smaller scale Boma
-Boatwrights: vegetarian jambalaya
-ESPN Club: veggie sandwich and cucumbler salad
-Coral Reef: veggie/potato strudel
Some of my favorite choices at Signatures:
-Citricos: tofu dish
-Jiko-chakalaka
-California Grill-flatbreads
