"In those days I, Daniel, was mourning three full weeks. I ate no pleasant food, no meat or wine came into my mouth, nor did I anoint myself at all, till three whole weeks were fulfilled. Daniel 10:2, 3
Basically, its going vegetarian for 40 days.
I have a slightly different take--
For some this just means meat (and possibly also fish) free eating. One can have very decadent vegetarian diet where you eat creamy soups, sweet sauces, lots of rich desserts, etc. Very easy to follow at Disney. There are a number of good threads on yummiest veggie foods already.
In my mind this kind of fast is more about eliminating the extras. Sure, you can be vegetarian and have a huge omelet and French Toast for breakfast, Mickey Bars and Dole Whips for snacks, Le Cellar's Cheese soup for lunch, and a big bowl of pasta with cream sauce for dinner, but is that following the intent of the diet?
It all depends on your perspective and only you know the right answer for you, but for a more intentional vegetarianism with a focus on eating the most simply prepared foods, I'd recommend:
--Fresh fruit stands at all the parks
--The baked potatoes at Liberty Square Market in Magic Kingdom
--pineapple spears at Aloha Isle in Magic Kingdom
--Miso or Udon soup at Yakitori House (though may contain fish)
--Vegetarian Chili at Columbia Harbor house, as well as the Lighthouse sandwich
--Most of the menu at Tangeriene Cafe in Morocco
--Fresh restaurant at the Dolphin for Breakfast or Lunch
Epcot will be the easiest place to eat--lots of choices at Sunshine Seasons and at the International Showcase.
For other kinds of vegetarianism the buffets are a great option, but if you are trying to avoid excess they may not be a good option for you.
At a sit down restaurant, you can always speak to the chef if nothing on the menu works for you (just do so in advance if you can).
Have a great time and enjoy!