Hi Kimberley!
It sounds like you have a great start to your vacation planning! A Disney vacation truly begins with that first walk down Main Street to Cinderella's Castle!
I think you've also got a great idea by placing your visits to Hollywood Studios and Animal Kingdom before and after your Magic Kingdom and Epcot days. Since Magic Kingdom and Epcot typically have the longest operating hours, you'll be having shorter days and some time to enjoy the pool or other activities during the days you aren't visiting those parks.
No worries on dining with a vegetarian! As a fellow veggie-only diner, I can tell you, without a doubt, that Walt Disney World is the easiest place for me to eat! All Disney restaurants can easily accommodate a vegetarian even if you don't see any options on the printed menu. On our last trip, our family visited the Yak & Yeti counter service restaurant. Since I didn't see anything available for me on the menu, I simply asked the cashier what my options were. She gave me a few choices, and I chose a really yummy Sweet & Sour vegetable entree.
Here are links to a couple of my other answers about Magic Kingdom
quick service options and one with
healthy options for both counter service and sit-down meals.
Hope this helps! Have a great trip!