Another vegetarian here. Seconding agamble's suggestions of the Mexican restaurants, specifically Rancho Del Zocalo (DL). They are VERY veggie friendly and most of their items are customizable to be vegetarian (or even vegan).
We also like Jolly Holliday on Main Street. The caprese sandwich is really good (comes with house chips) and they have really nice salads too. Desserts are also amazing!
For a snack, you can't go wrong with a Mickey Pretzel, or better yet, the Bavarian pretzel at Sonoma Terrace (DCA).
Award Weiners (DCA) has a portobello mushroom sandwich that comes with fries or fruit.
White Water Snacks (in the Grand Californian) is a nice place to get a rest from the parks. Their breakfast options are really good -- I like the breakfast burrito (ask for no bacon). For lunch/dinner, they have large pizzas that are enough to feed a family, we got the artichoke one and it was really good! I've also gotten their nachos without any meat added.
One thing I will not recommend is the jackfruit sandwich from Smokejumpers (DCA). We just had it for the first time a few weeks ago. It was very spicy and extremely messy to eat. If spicy is your thing, it may be a good option for you though.