Look at AllEars, they have a vegetarian dining section that lists the options at every restaurant.
http://allears.net/din/dining.htm under "Special Dietary Needs" in the left hand column.
AFAIK every table service restaurant has at least one vegetarian option although of course some are better than others (I mean, I've mostly heard good things about all of them, but for example at Cinderella's Round Table you'd get pasta with veggies, not exactly the most inventive vegetarian meal ever--whereas Garden Grill serves pumpkin risotto). You might want to call and get a notation put on your ADRs though. Most counter service places will have at least one option (usually veggie burger, veggie wrap, cheese or veg pizza, or some sort of Asian noodle thing/stir-fried veggies) although again, some are better than others. Off the top of my head: Tangierene Café and Sunshine Seasons in Epcot; Cosmic Ray's and Columbia Harbour House in MK; Starring Rolls in the Studios; Pizzafari in AK.
My husband and I are primarily vegetarian but eat some seafood and I think there was only 2 CS places that didn't serve anything we eat (Flame Tree BBQ in the Animal Kingdom and Cantina de San Ängel in Epcot). Obviously you will be more limited than we are, so it would pay to look at CS menus ahead of time to get a feel of what you can get where.
ETA: The vegetarian options at Crystal Palace look pretty amazing, AllEars says it's supposed to be the best character meal for vegetarians. We're not into characters so much so we decided not to do any character meals and let others have the fun, but I will definitely have to go there sometime!