Lots of good vegetarian food in WDW! There is a whole bunch of different threads and posts about it if you do a search, but I'll see if I can answer some of these.
Every trip I've ever taken to WDW has been as a vegetarian. I've had a great experience there regarding the food situation. Some of the restaurants I had a little apprehension with have often turned out to be the ones I enjoyed the most!
As far as CS each will offer at least one vegetarian option..
MK:
Cosmic Ray's - I love this place. Not only do they have veggie burgers with a great toppings bar, but they also have a veggie wrap.
Village Haus - They used to have a veggie pizza, but I no longer see that on the menu. They do have a cheese pizza, though. You can also ask that the Mediterranean salad be made without the chicken.
Columbia Harbour House - They have vegetarian chili and a hummus sandwich.
Main Street Bakery - I've read that they have a tomato and mozzarella sandwich, but I haven't tried it yet. I usually go there for their tasty cinnamon rolls.
Pecos Bill's - Veggie burger here.. Also you can order the taco salad meatless.
Epcot:
Sunshine Seasons - Love this! Okay, so a lot of options here like vegetarian sushi and a beet salad, but what I really love here is the tofu noodle bowl. They also have a veggie sandwich. Good, but that noodle bowl will keep me coming back.
CS in the World Showcase can be a little tricky but there are things out there.. Apparently, you can get pizza by the slice at the Via Napoli window. I haven't tried this myself so I can't comment on it. I recently read that someone was able to get a vegan meal from Lotus Blossom Cafe. Before you had to worry about chicken stock in their "vegetarian" offerings, so I've always steered clear of that place. We generally tend to go to Liberty Inn because I know that the veggie burger is in fact veggie.
AK:
Pizzafari - They have cheese and veggie pizzas here. I've read that you can get the antipasto salad ordered vegetarian.
Restaurantosaurus - Veggie burgers are offered here and they have a decent toppings bar.
Flametree BBQ - The fruit plate is really tasty! Really nice on a hot day.
DHS:
Okay, I have no suggestions here because I don't do this park, but I'm sure you can find cheese and/or veggie pizzas at Pizza Planet. And a veggie burger somewhere in the park.
Resort CS:
Polynesian - I love Capt. Cook's! They have an adult grilled cheese sandwich, a noodle dish that can be made vegetarian and veggie sushi. They also have cheese flatbread pizza. And in the morning tonga toast! Yum!
Wilderness Lodge - Roaring Fork - They have a veggie sandwich and cheese flatbread pizza. I know they also have a salad offering.
We will be trying Gasparilla Gas and Games this year. The menu sounds promising. I also know Contempo Cafe has a few veggie dishes that sound good as well.
Every TS will have at least one offering. And the buffets are actually pretty good! Some of the more surprising places I have found really good vegetarian food have been Boma, 'Ohana, Biergarten, Whispering Canyon, Liberty Tree Tavern and Cape May's Clambake Dinner.
If you are at a restaurant and you are worried about something.. Ask to see a chef. A lot of times they will gladly come to speak with you. And at most buffets a chef is usually happy to walk you through pointing out what you can and cannot eat.
I hope this has been a little helpful to you. I encourage you to take a look at some of the menus online and see what they have to offer..
http://allears.net/menu/menus.htm
http://www.wdwinfo.com/wdwinfo/dining/disneydining.cfm
If you have any other questions, feel free to ask! There are lots of vegetarians and vegans around here that are more than happy to answer your questions. 
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