What a great mom you are!
We are a vegetarian family (we eat dairy and eggs, although my DD is allergic to dairy so she doesn't eat it). We've taken kids between 1-8 to Disney several times and I look at the menus on AllEars to make sure I pick CS places that offer healthier food (I won't let them eat processed pizza everyday--yuck).
One thing I have learned is it is better to find CS locations that have no veg kid option but do have veg adult options. If you ask a mgr, they will always give the kids the adult option when that is the case.
Here are some better options:
MK
Pecos Bill offers a kid's hamburger but you can ask for the veggie burger instead (anywhere that has a kid's choice as hamburger but sells veggie burgers will sub for your DD, even if there are other veg choices--you may have to ask a mgr).
This would be OOP but for just $4 you can buy your DD vegetarian chili at Columbia Harbour House and just use the extra CS for breakfast or something.
Epcot
The Asian counter at Sunshine Seasons will give your DD a tofu, noodle, veg stiry-fry that is really good (you may have to have them ask a mgr because a lot of times they don't think they are allowed, but we always eat there and the mgrs never hesitate)
Tangierine Cafe has 2 awesome veg CS options but for adults. A falafal sandwich w/ sides of couscous and lentil salad or a veggie platter w/ falafals, hummus, couscous, olives, and lentil salad. Kid's pick is only meat there, so they will offer you a veggie kid's choice. I actually am planning to call and ask what they will let my kids have there.
DHS
Backlot Express-only meat kid's options and the mgrs have let my kids have the grilled veggie sandwich on 2 diff occasions.
AK
This is the most difficult place to eat veg.
As a PP said, there's the fruit platter w/ honey yogurt at Flame Tree BBQ (I think DD and I will just pair this with a soft pretzel and have to call it lunch this time--we have eve ADRs at Boma). I will have to get DD soy yogurt and make sure the brand they use doesn't contain gelatin.
There are some adult veg options you could pay OOP for or ask about. Restaurantosaurus has a veggie sub and Safari Eggrolls has vegetable eggrolls for about $2. You could buy an eggroll with some fruits and vegetables from the Harambe fruit market, and a soft pretzel (the pickings are slim here).
Sticking w/ buffets for TS is the best idea, and the chefs will always make things from scratch, too. They will always walk you through the buffet and point out what is veg, but occasionally you have to ask them to (some things look veg but are cooked in beef or chicken stock, etc). The best vegetarian-friendly buffet is Boma at AKL. Also Tusker House, Hollywood and Vine, Crystal Palace, and 1900 PF (if you can confirm that chef TJ is there, he will make your DD a gourmet veg meal for no extra charge--he is awesome). I have heard Ohanas will do an awesome veg entree if you request it (not sure if you have to call ahead).
I see you have an ADR at Akershus. The only veg dish is mushroom-stuffed pasta in cream sauce. My kids do not like this. Just wanted to let you know so you can know if your DD will eat it.
HTH!