There is actually a fairly simple set of principles to observe when seeking out decent veggie fare at the Mouse's house:
The least expensive restaurants' menus are pretty much set in stone. Some, like Cosmic Ray's, are pretty veg-friendly. Others, like Liberty Tree Tavern (at the Magic Kingdom), are almost veg-hostile; baked-potato territory. The discrepancy is actually part of a master plan called "itinerary planning." The idea is that every restaurant at WDW offers a slightly different, but complimentary, menu. "Don't feel like Mexican? The Dole Whip stand's coming right up!" And so on. The attitude extends to vegetarian offerings as well. That's one of the reasons we feel Vegetarian Walt Disney World and Greater Orlando is a really useful book: We've already found the veg-friendly places for you, and their menus, with a few exceptions, have not changed very much since we wrote the book.
Second-tier restaurants--by which we mean in-park dining rooms like Tony's Town Square restaurant (on Main Street in the Magic Kingdom) or Chefs de France--can be quite good, but again you have to know where to go. Our book, as well as the rec.arts.disney.parks Usenet newsgroup and sites like yours and Deb Wills' WDW Information Guide can be lifesavers in finding these spots. As a general rule of thumb, the inherent veg-friendliness of the restaurant's featured cuisine is a good indicator of your meal's quality as a vegetarian or vegan. Middle Eastern? Yup. Mexican? Go for it. German? Um...
By and large, the greatest kudos go to restaurants in the more expensive resorts. The Contemporary's California Grill, Jiko and Boma in the year-old Animal Kingdom Lodge, and of course the very pricey Victoria and Albert's, in the Grand Floridian, are all veggie standouts. Artist Point (at the Wilderness Lodge) was a haven for anyone with a special diet a couple of years ago when Anette Grecchi was behind the stove, but recent meals have been a little uninspired from our perspective--pretty much since Anette left! The good news is she's now cooking at Jiko.