I'm not a vegetarian, but I prefer vegetarian meals most of the time. I very rarely order a meal with meat and had absolutely no problem at any of the WDW restaurants over the past few years.
Almost every counter service I have ever been to offers nice salads. Some of them come with chicken or other meats, but I usually ask for them without the meat and have always been accomodated ( usualy at a reduced price!).
I shouldn't complain about WDW meals, because I know they are trying to accomodate lots of different people from all over the world, but I really find their meals full of meat, carbs and very unhealthy in general. Even the salad bars in the buffets seem to be full of heavy, high calorie foods, with lots of heavy dressings.
I'm from the Pacific NW and am lucky that we are so close to Calif. and have access to wonderful produce year-round. I never realized how healthy our restaurant's were until I went to Orlando. I think the Southern influence really shows in their menus and makes it hard to eat healthy. If you are not used to eating that way, it really takes its toll on your stomach.
But what the heck, I'm on vacation anyways, so I try not to worry too much!