Hi, Lisa!
Absolutely!
We never enter a park without a bag of some sort, from when we needed a diaper bag to later needing a bag for snacks and sippy cups and now needing a bag for water bottles, sunscreen, sunglasses and snacks. No worries!
Whatever you may need to bring with you can easily be brought onto every attraction in Walt Disney World, too, as each attraction has a compartment where you can stow your gear safely. Should you wish to leave your bag in a
locker, though, affordable lockers are available in each of the parks as well.
Each year, I bring hundreds of high school seniors to WDW for their Senior Class Trip; one of my responsibilities is to check in at First Aid, which is located in each of the parks. Should you need to seek the assistance of one of Disney's nurses, be sure to visit a
First Aid Center; they are super-friendly and really love to help Guests!
Send more questions as you think of them, Lisa; I'd love to help!