Kristen,
Thanks for asking your question to planDisney!
Many of the rides at Walt Disney World will have storage areas in the ride car seats that can hold moderately sized bags. But that is not going to be on all attractions, and the spaces will not always match your bag size.
But no worries! I have a great suggestion for you that will let you safely store your bag while allowing you to enjoy your park days. Your supplies will be nearby when you need them, but you don't have to worry about bag safety when you are zooming around on
Space Mountain.
Lockers! In each park there are
lockers available to rent. They are at the front of each park after passing through the turnstiles, and there are a few locker sizes available. They are keyless and typically will cost you $15 or less for the day. The
Magic Kingdom Park and
EPCOT even have a few larger lockers available that are up to 17" x 22" x 26" (43 cm x 55 cm x 66 cm). Prices can change, so check first online for current rates.
I think this is going to be your best bet. You will have your supplies within the park at easy access, but again, you don't have to constantly worry about where the bag sits on some rides.
I hope that this helps you have a better park experience.
Have a Magical Vacation!
Doug