Hi there, Douglas!
Each Park at Walt Disney World offers rental lockers, which are meant for a day's use. A
list of locations for the lockers may be found on Disney's official Web site.
The challenge with renting a locker is that if you want to store something you'll want or need repeatedly throughout the day - cell phone, camera - you will need to spend a great deal of time running back and forth between the locker area and the attraction you are currently visiting.
I've found that a small cross-body messenger bag or backpack affords both safety for valuables and freedom of movement. If I wear my cross-body bag to the Parks, when I am on a ride, I simply leave it on and negotiate the safety harness or seatbelt over it.
Other Guests swear by backpacks, which, depending on the contents and the size, can be kept on the shoulders or settled securely by the feet on rides. I know Guests who will go so far as to nestle the bag firmly between the feet, sliding one strap over each ankle! But footwells on the rides - including thrill rides tend to be deep enough that this is rarely necessary.
Additionally, many rides - particularly the thrill rides - offer seat pockets for smaller bags and valuables. These are often secured at the top by snaps, an added measure of protection for your things.
Make sure to do your souvenir shopping on the way out, though; those bags can become somewhat unwieldy, depending upon your purchases, and can be difficult to fit in provided space.
Have a great time!
See you later, celebrator,
-Deborah B.