Our family of four has one strict veggie, one part-veggie-will-eat-fish and one not-really-veggie-but-doesn't-like meat. We've never really had a problem finding something to suit us, even if it's meant relying quite heavily on salads and pizza every now and then!
Most restaurants will happily leave out the meat part of a dish if you ask.
WDW full service restaurants will provide a specially prepared vegetarian dish for you with 24 hours notice.
Sweet Tomatoes on ID was a big hit for all of us (carnivore DH included! )
Our favourite for veggie fare in WDW is at Boma's in AKL. The buffet includes loads of veggie dishes anyway but once, because I asked the server if she could point out all the items without meat, she asked the chef to cook something vegetarian especially for us as well - it was wonderful!
As I said, we've always managed to find something wherever we've eaten, but I'm struggling to come up with any restaurants that had particularly veggie-friendly menus. DD and I have both put on our thinking caps and I'll post again if I get any inspiration!
Boma
Jiko
Liberty Tree Tavern
Donald's Breakfastosaurus
Brown Derby
Blue Zoo
Fultons Crab House
Earl of Sandwich
Kona Cafe
Tangierine Cafe
Grand Floridian Cafe
Mythos
Citricos
Le Cellier only has one veggie starter and main - but they were both really nice. My very favourite veggie placce though is California Grill - perfect for that special night out!