I was just there in November and I am a very strict vegetarian. They did have veggie burgers everywhere that they served regular hamburgers. However, they were the garden patties, which I don't particularly like myself (I prefer soy patties).
I haven't heard anything about veggie dogs, though I know that Yves markets their veggie dogs to restaurants, since Hot Dog on a Stick carries their dogs to make their veggie corndogs with.. so Disney needs to get with it!
Counter service dining has been really difficult in the past, but lately they have at least one item (other than salad..yuck!) for vegetarians. They had mushroom sandwiches at a few places at AK and veggie sandwiches at a lot of other places throughout the resort. If you're sitting down for any meal, they'll pretty much make anything you want if they have the ingredients. However, I really enjoyed eating at both France and Italy. They had the best selections EVER for vegetarians! Everything was so yummy!