We have done loads of character meals and I am a vegetarian, I have alway's found there is something for me to eat.
I went to Ohanas last year and they cooked me something special..although I can't remember what it was LOL WDW restaurants are always willing to whip you something up.
I have done several character meals and have found that the only things available for me have been salads or pizza from the childrens section. I have always spoken to the chef and they have been really good. I usually ask for a fairly simple dish such as pasta alfredo and they can rustle it up fairly quickly. Another alternative would be to make a PS the day before and request a vegetarian option.
There are vegetarian options at many of the counter service restaurants. These are listed on the park maps. My favourite is the vegetarian chilli bread bowl available at the Columbia Harbour House in MK.
This link is useful WDW Dining.
Tracy