As a fellow vegetarian, I find eating at WDW much easier than many local restaurants in my hometown. WDW does an incredible job meeting the many, diverse dietary needs of guests.
At all the table service restaurants, you will be able to find a vegetarian offering. Similarly, nearly all of the counter service restaurants also offer a vegetarian choice. You can also request to see a list of ingredients if you have any concerns. Similarly, chefs will even come visit with you at the table service restaurants and may prepare something special for you. On our last family trip, I ate the entire week following a vegan diet and did so quite easily!
A few of my favorite choices for vegetarian dining are Bomas at the Animal Kingdom Lodge (this is hands-down our family favorite!), Columbia Harbour House at the Magic Kingdom, the California Grill at the Contemporary Resort, Sci-Fi Drive In and the Brown Derby at Disney's Hollywood Studios and Sunshine Seasons at Epcot. You will also find many fruit stands throughout all of the parks that offer yummy, healthful snacks, and soy milk is easy to obtain, too.
If you like more details about vegetarian dining and menu offerings , you can visit www.allearsnet.com and do a search for vegetarian dining.