Let's see, counter service at the places you named: (Veggie burgers are sadly inescapable, sorry)
MK:
Cosmic Rays has a veggie burger and a vegetarian wrap.
Pecos Bill's has a veggie burger, and a really tasty salad (has cheese in it, though, of concern to vegans.)
The Baked Potato Cart in Liberty Square, if it's open.
Epcot:
Tangierine Cafe in Morocco has a (very good) tabouleh wrap and also a vegetarian platter. (The sit down restaurant is worth a visit, too.)
The China counter service (name escapes me) has a veggie stir fry.
The food court in the Land has at least one option.
MGM:
The Commissary has a tabouleh wrap and a veggie noodle stir fry.
Backlot Express has a grilled vegetable sandwich.
The water parks: You're sadly out of luck here, as far as I know... I console myself with Rum Runners.
Downtown Disney:
Rainforest Cafe has a (really good) veggie burger, and a portobello mushroom sandwich... not counter service, though.
Puck Express at the marketplace has linguine with tomato and basil
Sit down meals: The higher-end places have some GREAT choices. Citricos and Jiko come to mind (Jiko will even replace some of the meats in their entrees with tofu if you ask.) Kona Cafe at the Poly has a great veggie stir-fry. Generally speaking, WDW chefs are amazingly accomodating at this kind of thing, I've frequently had a chef come out to talk to me personally, or had the server produce a notebook with a complete ingredient listing as soon as I start asking vegan-style questions (which, honestly, is a little more attention than I was hoping for
but I can't complain about the effort.)