Pamela,
Disney World is improving their performance in the area of vegetarian dining. That is not to say that they are a vegetarian travelers dream yet, but they certainly are trying, and you should be able to find lots of good, if not great, options. I think you will find that many of the restaurants have some kind of vegetarian choice, whether they be a table service or counter service location. But not all of them do, so do a bit of research.
First, there is a great series of articles and links at AllEars.Net. These can help you identify where and what to eat in each park, and each resort. They even go into Vegan/Vegetarian distinctions, and a few articles chronicle some past guest experiences.
Second, look at the menus ahead of time. While I have never shown more than a passing interest in this matter, I quickly found on the dinner menu at
Shutters (table service dining at your resort) that they have a vegetarian version of the Caribbean Pasta dish, and one fish dish (which for some vegetarians is considered OK).
I hope that this helps.
Have a magical (and delicious) vacation!