For MK...
CS - Columbia Harbor House - has a great Lighthouse sandwich with hummus, broccoli slaw, and tomato. They also have a veggie chili.
TS - Crystal Palace has a nice buffet with a surprising amount of veggie offerings (various salads, pastas, rices, veggies, sometimes soups, etc).
For AK...
CS - Not a ton of veggie options, which is surprising considering...but Restaurantosaurus has a veggie burger and a toppings bar, so you can load it up with whatever you like.
TS - Tusker House has a good amount of veggie options, but if you don't mind making the trip to AKL, Boma is even better (IMO).
For DHS...
CS - Again, nothing to impressive. Backlot Express has a veggie sandwich with black bean salad. Nothing to rave about, but a nice little change from the regular veggie burger. Pizza Planet also has a plain cheese and a veggie pizza that comes with a small side salad.
TS - I like 50's Prime Time - stuffed pepper filled with brown rice and ratatouille. They also have DELICIOUS PB&J milkshakes.
For Epcot...
CS - I like the Cantina in Mexico. Although they don't have much (if anything?) vegetarian on the menu, they'll make you tacos, burritos, etc. with what they have on hand. It takes a few more minutes, but in my experience, totally worth it. I also hear great things about Morrocco and Sunshine Seasons...definitely stop by the bakery in France for a treat.
TS - I really like Akershus. At lunch they have a delicious grilled veggie pasta, in addition to a cold buffet w/ salads, cheeses, etc. They're very accommodating if you'd like something different though. Veggie lasagna at Chefs de France is also very good.
If you're going to a buffet, you may want to note Vegetarian with your ADR or call the restaurant ahead of time. Some things may be prepared with chicken broth or cooked with meats, and a chef can tell you what you can/can't eat. Have fun!!