O- I know your dilemma- I had to plan a trip with people in the exact same situation! For the vegetarians in your party head on over to the Tusker House at Animal Kingdom (the vegetarians in my party raved about that pick!), Boma at AKL, either restaurant in Italy at Epcot, as well as the Moroccan Pavilion at EPCOT. And, although we did not eat there, the Mexican Pavilion at EPCOT also had vegetarian options. Also, most restaurants were very vegetarian friendly when it came to picking a place for Breakfast, there were almost too many options!
For gluten-free it is a bit trickier but it can be done. The Japanese Pavilion at EPCOT was very accommodating at all their restaurants, along with the Chinese Pavilion. Also we found many good salads and gluten-free options along the way at many quick-service stands.
One thing I did discover, even if they did not offer many of the specifics that we were looking for, was that most restaurants at WDW were very accommodating to our dietary restrictions. I simply told them ahead of time, when we made our ADRs what our needs were, and reconfirmed them once we arrived to check in. In many cases the chef came out to discuss our options with us, and in some instances even made some special items like gluten-free cakes, or a special vegetarian dish taking us all by surprise!
