The restaurants will work with you for your dietary preferences. This menu will also be helpful.
http://www.allearsnet.com/menu/menus.htm
I have heard that restaurants at Animal Kingdom offer vegetarian meals on their menu. I rarely eat meat and never had a problem with any of the menus.
My children are smaller but had great birthday celebrations. For my son's 4th birthday we had breakfast at Crystal Palace and rode the monorail to the Grand Floridian for the character dinner at 1900 Park Fare. It was a nice setting, everyone enjoyed the atmosphere and they made a wonderful Mickey birthday cake with a cream filling and ganache topping... it was YUMMY.
We celebrated our daughter's 1st birthday at Cinderella's Royal Table for breakfast. It, too, was great.
I would ask your daughter to pick a theme park and go over a list of restaurants with her, restaurants in the park or surrounding resorts. If she would like character dining this link will help:
http://www.kingdommagictravel.com/disneyworld/dining/character_dining.htm
Chef Mickey's has the best reputation for celebrating birthday in Birnbaum's book. I would purchase a 2007 copy, talk with a Disney travel specialist and do research online. We surprised DH and celebrated birthday at Chef mickey's last May. That is is accessible by monorail from the MK and Epcot.
Whichever restaurant you choose be sure to mention at time of reservation, and upon arrival, that you are celebrating a birthday, and get a birthday pin from the park.
I love celebrating birthdays at Disney. Everyone feels special when they get the extra attention, birthday cards/certificates from the characters and cast members notice the birthday pins... it makes a wonderful birthday memory.
Also, the resort may do something as well. It's worth a mention. I decorated our room since the birthdays were during our stays.
Have a wonderful time planning and be sure to throw in a few unexpected surprises if you can!