My DD is a vegetarian. Every TS restaurant will have at least one vegetarian option. It's hit and miss at the CS restaurants.
Check out All Ears Menus here for current menus
http://www.allears.net/dining but be prepared for change.
Magic Kingdom
Be Our Guest with 2 vegetarian options, plus a tuna nicoise salad
Columbia Harbor House with the Lighthouse hummus sandwich
Pecos Bill's vegetable rice bowl
Epcot
Sunshine Seasons has a few vegetarian options, plus a sesame crusted tuna asian noodle salad
France, Boulangerie Patisserie les Halles has vegetarian quiches plus yummy pastries
Japan, Katsura Grill has Shrimp Udon. I'm not sure what the broth is made from, we "don't ask, don't tell" because meat is a textural thing with my DD. She removed the shrimp and eats the noodle soup.
Animal Kingdom
Harambe Market used to have the BEST vegetable stack to die for, now it has veggies over rice with yogurt raita. I'm bummed.
Satuli Canteen has
delicious tofu bowls
Mr Kamal's is all vegetarian with falafel and hummus
Hollywood Studios
Is a wasteland of good food for vegetarians and carnivores alike
Backlot Express has a caprese sandwich that isn't too horrible. Not great, but not horrible either.
ABC Commissary has a veggie burger (one of the last ones left in WDW) but I havn't tried it because we've sworn off of eating there. We will probably stop by when my DD joins is next so she has something to eat.