I'm a strict lacto-ovo vegetarian and my 20 month old DD is also a vegetarian. DD also has a severe egg allergy.
Being a vegetarian at WDW is very easy (vegan is a little tougher!

). One of my favorite things about our WDW trips is that I don't have to worry about explaining my vegetarianism in hope that I might be able to find a little something to eat. At TS restaurants, there is always at least one vegetarian entree. The chef's always went above and beyond to accomodate my DD's dietary needs, even though she is under 3 and was free in several restaurants we went to! Garden Grill prepared a special meal for her (egg free pasta, instead of the normal veggie ravioli, topped with a tomato base sauce and veggies). They also brought her out an Itzakadoozie for her dessert.
I second the Boma recommendation. I am not a buffet fan at all at home, but Boma is a must do restaurant at Disney for me. Your server will ask you if anyone in your party has any dietary concerns. They have always brought a chef out to walk me through the line. One chef told me that 70% of the food at Boma is vegetarian. The last time we were there in January, the chef I spoke with brought out a special soup for me (it was one I liked that wasn't out that night), extra food for my DD, and a special dessert for DD.
Other places we have been happy with...
TS: Crystal Palace, Kona Cafe, Le Cellier (crazy, but they have an excellent vegetarian entree), Coral Reef
CS: Tangerine Cafe, Columbia Harbour House (they have veggie chili and a good vegetarian, but not vegan, sandwich).