Hi Kristin,
I hope this question reaches you prior to your arrival at the Walt Disney World Resorts in a few weeks!
The only restaurants that would have access to salad bars would be the buffets. All other restaurants would serve individual portions of salads to guests. I would recommend you visit 1900 Park Fare in the Grand Floridian. This salad bar is small but of higher quality. It is a character meal so you will pay slightly more money and it is my favorite character meal. They may even have strawberry soup on the menu which is delicious.
Chef Mickey's has a salad bar but it is okay and this restaurant is a character meal as well. The other place that might work for you is Boma in Animal Kingdom Lodge. This is one of may favorite restaurants and it is a buffet that often has quinoa on the menu.
Your best bet might be at Disney's Art of Animation's
Landscape of Flavors. Though this is a quick service restaurant, only the speed of service is was makes it fall into this category. They have really unique flavors from a variety of cultures. They even offer a make your own salad option on the menu.
While I am not a vegetarian, I do order many vegetarian options when I dine at the Walt Disney World Resorts. I have had some of the most delicious meals ever at Whispering Canyon Cafe, Yachtman Steakhouse and the California Grill and they were all vegetarian.
Wishing you a magical vacation!
Beth