There is a book about vegetarian restaurants in WDW. I don't remember the title, but I do own it, and it's a very good book. I'll dig it up tonight if I get a chance. I'm not a vegetarian, but I won the book in a contest.
Just to clarify: you don't eat fish? It's a bit hard to tell from your message. In any case, for vegetarian meals, it's probably better to have a big meal (I'd recommend lunch over dinner) at a sit-down restaurant, and a snack later in the day. Italian restaurants are usually good bets, since pasta and pizza can be done very well without meat. Most (if not all) locations in WDW will have at least one vegetarian entree, but these are more dependable bets, IMHO. Unless noted, these are sit-down restaurants.
Magic Kingdom:
Tony's Town Square- Italian
El Pirata y el Perico-fast food, tex-mex style
Epcot:
The Mexican fast-food place has lots of bean-based dishes. I'm sure the sit-down restaurant also does, but I've never been there, so I'm not sure
Boulangerie Patisserie: French pavillion, fast food. While the special here is definitely the to-die-for desserts, they do have some sandwiches and quiche.
Pasta Piazza Ristorante: In Future World. Fast-food Italian. Not great, but not bad either
Pure and Simple: in the Wonders of Life Pavillion, fast food. A lot (for WDW, relatively

) of healthy selections.
Check out the Sunshine Season Food Fair at the Land Pavillion. Go for a late lunch, as it's very crowded during prime lunch hours.
MGM Studios:
Mama Melrose's- Italian. Pizzas and pastas.
ABC Commissary: American (kind of California style) fast food
Toy Story Pizza Planet: fast food pizza. WDW fast-food pizza isn't usually that good at all, but it's better here than anywhere else, I think
Animal Kingdom:
Pizzafari- Italian fast-food
**Just a side note. I find it ironic that so many of the food places in the Asia and Africa areas of the park have so many meat-based food places, pretending it's authentic food, when so much of the diets of people in (the real, non-WDW) Asia and Africa are often vegetarian, and meat is not served every day!
Resorts and other outside of the parks locations:
California Grill at the Contemporary-great place!
Wolfgang Puck-the restaurant (Downtown Disney West Side) and the express location (at the Marketplace)
Boardwalk Bakery: some sandwiches available, I think. Not worth a trip to the Boardwalk just for this, but if you're going there anyway...