I think for the most part *all* of them have some sort of options. It just might not be exciting or taste good.
Two exceptions:
China at Epcot - they list foods as vegetarian that use chicken broth or fish paste. They weren't very accommodating either when I asked for something made with no meat, fish, chicken/beef broth or fish paste. I ended up with a slimy plate of gross vegetables. They were seriously slimy, and I'm not sure why.
Yak and Yeti at AK - They didn't really have any dishes you could even leave meat out of and still be vegetarian except a salad. That wouldn't be so bad except when I arrived I asked someone who appeared to be in a management position if there was anything vegetarian and she said, yes that I could get the Lo Mein without the chicken, and I asked if it was made with chicken broth, and she said no of course it wasn't. I went to order it and the server asked me if I wanted it without chicken because I was vegetarian, and when I said yes, he said that I shouldn't order it then because it was made with chicken broth. After the person I had asked at the front of the restaurant had told me specifically that the Lo Mein was not made with chicken broth. He checked with the chef for me and I pretty much had to get the mandarin chicken salad with no chicken because that was all that could even be made vegetarian.
If you want to eat somewhere with a *lot* of vegetarian choices, I think Boma at the Animal Kingdom Lodge is the only place I can think of where you get more than one or two choices.
I remember the Crystal Palace buffet also having a decent number of choices as well.
I haven't eaten there since the conversion to buffet, but I've heard the Tusker House at AK has a decent selection of vegetarian options too.
Columbia Harbour House at MK - this is counter service, so not sure if it's what you're looking for, and again, they don't have a lot of options, but they do have a good vegetarian chili and a hummus/brocolli slaw sandwich that's pretty tasty as well
In the past, some of my favorite meals have been at restaurants that had nothing vegetarian on the menu at all, and I just asked the server what I could get that was vegetarian.
1) O'hana's at the Polynesian - It wasn't the best the last time I went, but previous times, the chef made me really delicious tofu/noodle dish with veggies. It was the same thing last time I went, but it wasn't prepared nearly as well and was super salty and just didn't taste good.
2) San Angel Inn at Epcot - If you ask the server, they have a small vegetarian menu they can bring you with basically two things on it. Not so good if you don't like those things, but great if you do like them. I think they're really tasty personally:
http://www.disboards.com/showthread.php?t=1607071
I emailed Disney guest relations a while back about vegetarian options for a specific restaurant I'd never been to, and this is their response:
We appreciate your interest in learning what accommodations are
available in our theme park and resort restaurants for our vegetarian
guests. Please know that vegetarian plates are available at many
counter service locations and at all full-service restaurants, even if
they are not actually featured on the menus. Also, our chefs and
restaurant managers are always happy to provide information regarding
not only the ingredients of menu selections, but also how items are
prepared. In many instances, food may be specially-prepared to
accommodate particular dietary restrictions.
