Hi, Angela! I hope you will be pleasantly surprised at how many options there are available to you for dining at Walt Disney World with a peanut allergy. There are many snack options that are peanut free, and they include everything from fresh fruit to cookies and brownies. You can ask at any location, and you'll find prepackaged items from manufacturers like Enjoy Life and Surf Sweets. The cast members are also pretty well versed at helping you to identify things that are safe, even when it comes to snacks.
You will find quick service dining locations are very familiar with allergies. I love Cosmic Ray's at the Magic Kingdom, where you'll find options for just about every palate. Sunshine Seasons Food Court at EPCOT is another great dining location. Again, rely on the cast members working at the restaurants to provide you very specific information about what you'll be able to enjoy. Menu items and ingredients are subject to change, so you want to make sure to check when you arrive for the best information.
A website you might want to check out is Allergy Free Mouse. Dealing in detail with special dietary needs resulting from allergies, this is real life experience from people who have enjoyed Disney dining many times with allergies such as yours.
I also want to recommend that you visit
Walt Disney World Special Dietary Requirements to give you even more information.
Have a great, peanut free vacation!