Hi Moriah,
Good news! You can take backpacks, purses, string bags and other totes with you on every ride and attraction at Walt Disney World. Usually, there is space to place your bag on the floor or the seat beside you. On some rides, there is a pocket in front of you that can be used to secure your bag.
I usually carry a medium-sized cross body bag and normally don't need to remove it from around my neck while on a ride. My husband carries either a regular backpack or a string bag and never has an issue finding space for either.
I hope that eases your mind. My niece has a peanut allergy, and I know how important it is to have an epi-pen handy at all times for safety and peace of mind.
By the way, Walt Disney World goes out of its way to accommodate guests with conditions such as your son's. Make sure you let the restaurants know about his allergy when you check in. Usually, at the table service restaurants, a chef will be available to come to out and talk about which options will work best. At the buffets, they will even point out which dishes should be avoided.
You can find more information about
special dietary requests at this link.
Thanks for visiting the Disney Parks Moms Panel with your question, and please let me know if I can do anything else to help you have an amazing first visit to the Walt Disney World resort. Enjoy!
Wilma