Watch out-- my whole family slowly converted to being vegetarians one-by-one after my 7-year-old sister made that same choice 14 years ago!! (This had a lot to do with the on-going story of Tom the Turkey that she wrote, revised, and read to us each and every Thanksgiving. Direct quote: "His name was Tom. He had a family." Yeah. My parents were thrilled. To make matters worse, neither she nor my youngest sister will even so much as look at zucchini or mushrooms. IT'S NOT EASY TO FEED THEM. But it is easy at Disney, where there is plenty of pasta, fries, and pizza to go around.
I've done a lot of looking into vegetarian food at the restaurants, and the best value for the money--especially for vegetarian kids (although she'll be above the kid's menu age)--always seems to be the buffets, because you can always try just a small taste of something (anything to get just a little nutrition onto that plate) and there are always plenty of vegetarian options available. But if she's happy with the basics (including mac and cheese or other pastas), you'll be safe just about anywhere. Even places that don't have veggie items on the menu (e.g. Yak and Yeti Counter Service or Hacienda de San Angel) will either have items available on request, or even a whole separate vegetarian menu with a couple options.
Have a great time! I'd be so interested to hear how it all turns out!