There are veggie wraps or grilled sandwiches at nearly every CS location. A few others have pizza, and some do have only salads, but they are definitely the minority and easy to avoid. Veggie burgers are available at any CS that has regular burgers, even if you don't see it on the menu, just ask. There are also several locations with stir-fried vegetables, vegetable salads, etc.
I see vegetarian options on every TS menu, with some restaurants actually having more than one, or another that can be easily made vegetarian. These entrees are primarily pasta, and some are tofu. But they are very different, from one location to another, so there is variety. Many restaurants also have truly vegetarian soups (no meat or fish stock), ask the chef to be sure.
Disney chefs are amazingly accommodating and truthful about ingredients. (I've found chefs in other places who could not comprehend that chicken stock is not vegetarian.)
Check menus on AllEars, here on the DIS, and on the WDW site, so you can choose your CS and TS locations with options you like.