Animal Kingdom:
There are no unique options here unfortunately. Cheese pizza or veggie burger only.
There is a cart between Africa and Asia that sells vegetable spring rolls. Not a meal, but at least a good snack or appetizer. Also, there is a CS place in Asia that has a Mandarin chicken salad. You can just ask them to put the chicken on the side.
Epcot:
SO many choices!
Sunshine Seasons
Tangierine Cafe
Sunshine Seasons is probably the best CS place at WDW for vegetarians. It is the only place I know of that serves tofu. Does anyone know of any others? You can order a meal of sides, like at the Asian counter, rather than sweet and sour chicken with two sides, you can just get 3 sides. They also have veggie sushi in the refrigerated case along with various salads and fruit offerings.
As for Tangerine Cafe, we actually ate there last night for the very first time. DD got a falafel wrap. Unfortunately, she really didn't care for it. Having tasted it myself, I have to agree with her. I love falafel and have had it many different places (including in Israel) and this just wasn't very good. It was dry and overly spicy. She ate most of it anyway but it isn't something she would order again. That's just personal preference, though, not Disney's fault (though I think they could make much better falafel).
If she really insists on eating off the child's menu her only will options are likely to be mac & cheese, pb&j, and cheese pizza.
This was part of the point I was trying to make. We've been to numerous CS places that don't offer any of these options. Quite a few places have one veggie option on the adult menu and that's it. Nothing to cater to the kids.
What does she eat at home?
All kinds of stuff. Definitely tofu and other soy-based items including prepared foods like veggie burgers and other meat substitutes like vegetarian chicken patties and nuggets, veggie ground beef, vegetable dumplings (both store-bought and homemade), tofu noodle bowls (homemade), sandwiches with cheese and veggies, vegetable stir fries over rice or noodles, pasta with veggies, etc. Just tonight, I made dinner in our condo at Windsor Palms. I sauteed cubed tofu, added a can of Chinese vegetables and served it over brown rice. Also microwaved some veggie potstickers as an appetizer. Cheap, easy and healthy (probably too much sodium in the canned veggies but otherwise healthy).