It is a good idea to tell them when you make your PS that you are a vegetarian, but sometimes they don't do anything in advanced and you still have to tell them when you check in. Also sometimes if you ask the waitor what is good for vegetarians they can help you or go tell the chef.
At buffets sometimes the vegetables, potato dishes, etc look safe but will contain chicken broth, so always asks the chefs behind the counter what is in the food to be sure. I know this was the case at Chef Mickeys' and Crystal Palace. At Crystal Palace I was brought out two pasta dishes as well as being able to eat the safe stuff from the buffet. At Chef Mickey's I was supposed to get something made special but I never got anything.
Garden Grille had a vegetarian dish that was really good. The Norway Buffet Resaurant Arkeshus is good also (but expensive considering you don't get dessert or drinks).
Rainforest Cafe has a vegie burger and pasta dishes.
I have heard Boma's is a good buffet for vegetarians but have never eaten there, same with Jiko. A lot of sitdown restaurants won't advertise they have a vegetarian option but I have heard many do if you ask.
From what I remember seeing for Counter Service there is :
MK
The sandwhich with hummus at Columbia Harbor House
A jicama salad at Pecos Bills
Nachos at El Pirata y el Perico (ask for without chili)
Vegie burger at Cosmic Rays
Vegie salad at Pinochio's Village Haus
Epcot
Pasta, potatos, fruit, pastry items at the Sunshine Food Fair in the Land
Nachos, or taco salad at the Cantina in Mexico (ask for without meat)
A vegetable Lo Mein that I was told was safe at Lotus Blossom Cafe in China Tangierine Cafe in Morrocco has a vegie platter with couscous, hummus, lentils, tabouhli, bread, olive salad.
MGM
Vegie or cheese pizza at Toy Story Pizza Planet
cheese pizza, fruit, pretzels, potatos, vegie burger at the food area by Tower or Terror (Rosies, Cantalina Eddies, etc)
Animal Kingdom
Tusker House has either a vegie sandwhich or a vegie wrap that has changed from time to time.
Pizzafari has cheese or vegie pizza
Flame Tree has kids PB & J deal (along with many other restaurants on WDW property)
Restaurantosaurus has a vegetarian salad
Food Courts at many of the resorts are okay but I have found really don't offer much for vegetarians besides pizza and pasta (And even then I have found beef in the sauce since ladels get drug on top of meatballs, so be careful), but many of them have sitdown restaurants that are probably accommadating.