The rice may be made with chicken stock, for example.
Vegan here, and one of my favorite places is Tusker House. Awesome buffet. Great for vegans, vegetarians, and omnivores.
An obvious vegetarian choice is Italian food. I like Mama Melrose's at DHS. Have never tried the Italian restaurants at Epcot, but I hear good things.
QS is harder and is getting harder still, but Tangierine Cafe at Morocco in Epcot has a great veg platter. Beautiful area, btw. Very pretty place to sit and watch the Epcot crowd. There are also some vegetarian choices at Electric Umbrella and Sunshine Seasons (both at Epcot).
At DHS, QS is probably PizzeRizzo--but beware, when I was there last they closed around 4 pm. So not for dinner. I couldn't find any other vegan QS options at DHS, but I'm not as familiar with the vegetarian options there.
At AK: Pizzafari is my QS of choice (because they'll make a vegan pizza). The QS restaurant near Dinosaur has a veggie burger, perhaps only on request. There are food carts here and there in AK that have different veg foods as well. Check out the lists on the WDW site. I haven't eaten at the new QS place in Pandora, where there are apparently many veg choices.
At MK: I've kind of given up. There used to be a fantastic sandwich at Columbia Harbour House, but they recently took it off the menu. If you like chili, they have that. There's some kind of falafel sandwich at Cosmic Ray's, but it's not vegan so I haven't had it and also I've heard it's not all that. Pecos Bill's might have some veg options.
Jiko at AKL has a vegetarian menu, and the food's great. Ditto Sanaa, although there aren't as many vegan choices there as there used to be and also much of their food is too spicy for me. YMMV. Boma at AKL is also excellent for everyone.
Lots of vegetarian places to eat at Disney Springs as well. Blaze Pizza is my go-to choice. And there's a vegan bakery at DS (Erin McKenna's) where everything's not only vegan, it's gluten-free.
Many sit-down restaurants at WDW have one vegetarian item on the menu and that's it. Even though every TS place will accommodate a vegetarian or vegan, that doesn't mean that the food will be what the vegetarian would like to eat. Sometimes the chef will create something awesome and sometimes it's just okay. I prefer to go somewhere where there's something already on the menu that I'd choose to eat.
And of course popcorn and soft pretzels are not just vegetarian, they're vegan.