Table service restaurants at Walt Disney World are a vegetarian's dream. My experiences have ranged from "good" to "outstanding". As a vegetarian, I never had any trouble finding something appealing either already on the menu, or available simply by omitting an ingredient or two. However, as a vegan, all of my best meals were items *not* found on the menu, but specially offered up when I mentioned my situation to the staff.
The following chart will give you some ideas of the more "veggie friendly" options, but keep in mind that each experience is unique. Mentioning your dietary needs when you make priority seatings will increase your options. I encourage every vegetarian and vegan to "speak up", and be prepared to eat well. Any full-service restaurant is happy to accommodate your special dietary needs - just give them a call ahead of time.