DD and I are both vegetarians. Every TS restaurant has at least one vegetarian entree on there menu. My favorite vegetarian meals on our last trip were Boma, Le Cellier (crazy since it's a steakhouse!), Kona Cafe, Coral Reef, and Crystal Palace. I would avoid Princess Storybook Dinner. The vegetarian pasta was horrible.
My favorite counter service restaurants are Columbia Harbour House and Tangerine Cafe. Columbia Harbour House has a wonderful hummus sandwich and vegetarian chili. Tangerine Cafe has a yummy vegetarian plate with middleeastern dips.