Hi Krista!
I'm so happy to be able to tell you that you don't need to worry at all!
Disney takes food allergies very seriously and a Manager or educated Cast Member at each food service location will be able to advise you on safe, peanut/tree nut free options to put your mind at ease!
I myself just traveled in December for 12 days at Walt Disney World and on a Disney Cruise with our good friends and their daughter who has a severe nut allergy. Every time we dined at table service or quick service locations - or even picked up a treat or snack, a friendly & knowledgeable Cast Member or Manager was always available to provide advice on which options were safe for her to enjoy. She enjoyed every meal!
Each time you're going to order, simply mention the allergy and ask about 'safe' choices for your child.
You can also read some information about
Disney's Special Dietary Requests via this link provided which also includes a listing for all Four Parks' Quick Service Dining Locations that are particularly equipped to handle allergies with specially prepared menu items.
You're going to have a Magical time - and your child will enjoy plenty of deliciously safe food options with you!
Kirsten