I'm a veggie and don't have too much trouble grabbing food at WDW. In fact most of the TS places have at least one option as a starter and entree.
But as to specifics:
TS
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Sanaa - Absolutely love this place. Great veggie options and the bread service is terrific.
Tokyo Dining - Most people associate sushi with fish but they do a great selection that is totally vegetarian (and maybe even vegan) friendly.
Boma Breakfast - It's a buffet but great for breakfast for veggies. If you like spicy I can recommend the Chakalaka.
Restaurant Marakesh - A good selection for veggies and the chef when we were there was very attentive to special food needs.
The Plaza - Is under rated but does a good selection of American fare.
And a honorable mention to:
V&As - If you really want to go up market you cant beat them. As your menu is designed just for you, you can't help but get amazing food. The only reason it's not in the top 5 is due to the cost but if you are a foodie it's a must.
CS
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The options are a little smaller here, usually with only one or two veggie options.
Tortouga Tavern - Opposite PotC, it was open seasonally when we were last there but when open they do a great veggie chilli.
Rosies - Almost a kiosk but a great burger for on the run. It's our usual lunch stop when in MGM *cough* I mean DHS.
Earl of Sandwich - At Disney springs rather than a park or resort but they do make fine sandwiches. Then if you fancy something sweet you can pop to Ghirardelli.
Restaurantasaurous - It's good ol' veggie burgers again but you can't beat a fixin's bar!
Sunshine Seasons - With all the different stations you're bound to find something here.
Honorable mention:
Cosmic Ray's Starlight Cafe - We've had veggie burgers, veggie wraps, and veggie chilli here (from different menus, I don't think they do all three at the same time) and who doesn't like being serenaded by Sunny Eclipse while eating?
But they are my favourites, I'd love to see a video done by the guys covering this (I could even help if they wait till Feb
)