I think one of the hardest things to wrap your mind around before you see it in person is just HOW BIG Disney World is. I read somewhere that "it is about 50 square miles....bigger than Manhattan and roughly the size of the city of San Francisco". Now mind you, I'm not positive that statement is accurate, but you get the idea. You probably wouldn't want to travel from one side of San Francisco or Manhattan each day just to eat some fast food.... and I doubt you will want to do it at WDW either.
The food courts at the resorts are no more or less homey/comfortable than the counter service locations in the parks. In fact some of them are more generic and LESS themed than some of their counterparts inside the parks. I think that you could just as easily STAY in the park and find something for everyone in your party to eat even if it meant stopping at a couple of different counter service locations to please everyone. For example, say a couple of you think the vegetarian options at Columbia Harbor House in the Magic Kingdom sound OK, but the rest of you would rather eat something somewhere else. It would take less time to stop and eat at BOTH places than it would to:
1) Make your way back to the park entrance. (Each park is large and spread out- it can take up to 15 minutes just to get from one side of the park to the other walking at a quick pace.)
2) Catch the tram to your car if driving, because even the parking lots are huge and usually your car is not within walking distance, or trek your way from the entrance to your Resort Bus Stop if using Disney transportation.
3) Find your way back to your Resort on the maze of Disney Freeways (you truly do leave the rest of the world behind and get up on Disney freeways) or W -A-I-T at the bus stop if using Disney busses.
4) Trek from the parking lot or bus stop into the Resort food court and then wait in the various lines for everyone to get their food, pay, find a table and eat.
5) Make your way BACK to a park to resume your regularly sheduled vacation.
Even more importantly though, in case you don't know, Disney World works very hard to cater to the dietary needs of their customers. You won't be the first vegetarians they've served, and pretty much EVERY place will help you modify their menu items to better suit your eating habits.
Although if at all possible budget wise, you might want to try to work in at least one or two table service dinners. The chefs at the sit down restaurants will talk to you and sometimes even create entirely new dishes for you based on your likes and dislikes. Maybe shoot for a nice meal on your first night, middle and/or last nights, with "questionable, play it by ear and hope for the best" counter service dinners on all the other nights?