There are vegetarian options at almost all QS locations. My favourite places to eat are these:
MK: Colombia Harbour House, Lighthouse sandwich mentioned above is amazing! The vegetarian chilli is pretty good too. Be Our Guest for lunch: Quinoa salad is so good! Pecos and Bill has a pretty good veggie burger. Oh, and the nutella and fruit waffle sandwich at Sleepy Hollow counts as lunch, right?
EP: Sunshine Seasons food court has tons of vegetarian options from sushi, to vegetarian tamerind beef, to amazing veggie flatbread sandwiches. My favourite QS for vegetarians in the world pavilions is Tangierine Cafe: get the vegetarian platter with couscous, hummus and tabouleh or the falafel wrap which is quite good. My next choice is at Les Halles Boulangerie Patisserie where you can get Brie aux Pommes, or Quiche Florentine, or Tartine Provencales, not to mention all the yummy pastries to enjoy afterwards. I have also eaten at La Cantina de San Angel which offers guacamole with chips, and cheese empanadas which are ok but nothing to write home about.
HS: The backlot express has a pretty good grilled vegetable sandwich and you can get a veggie pizza at Toy Story Pizza.
AK: Too bad he is vegetarian because that BBQ meat at Flame tree BBQ is so good, but don't panic there are veggie options available in the park too. Pizzafari has veggie pizzas. Restaurantosaurus has a black bean burger and Mr. Kamal's vegetarian cart has tons of veggie options: falafel sandwiches, asian noodle salad, hummus with pita and veggies, samosas, and edamame.
Hope that help!