First of all I want to tell you Disney is GREAT with taking care of vegetarians. My husband is a vegan and eats very well, so your daughter will too. At buffets the chef will come out and take her around letting her know what she can eat with her restrictions, and perhaps prepare something else for her.
I am focusing on table service as that is what you seem to want advice for. I have not eaten at many table service, but listen to liners. I tend to do mostly quick service or out of parks.
I suggest you pick through the restaurants at the parks and choose 3 for each park, print up the menus and let them pick from there. Also pick a few resort restaurants and again print menus and let them choose.
Oh and as to kid menus, anyone even adults can order off the kids menu.
At this age if you want a character meal, if they are not overly into it, I would pick Trattoria al Forno breakfast with Price Eric, Ariel, Rapunzel, and Flynn Rider.
At Hollywood Studios there is a lot who say Prime Time Cafe is good, it has a lot of comfort food and has some pretty awesome milk shakes.
Epcot - I have not eaten in many places, but hear a lot for Teppan Edo. There is Via Napoli which sees a lot of love.
Animal Kingdom, I have never eaten there. Tiffins has been getting a lot of new love, as well as Yak & Yeti.
Magic Kingdom I like Crystal Palace, but Be Our Guest is a hard to get favorite.
As far as resorts, I love Trail’s End at Fort Wilderness, great buffet, comfort food, awesome strawberry shortcake. And yes a vegan eats very well here.
Breakfast - Boma is my favorite.
Character - 1900 Park Fair Dinner
Overall I feel as if lunch menus are similar to dinner menus, but lunch seems cheaper.
This a bit dated but there is a lot of good here:
Again old, but my day one talks about my husbands meal at Trail’s End
http://forum.touringplans.com/t/my-trip-with-oo-de-lally/17667