Hi Melissa,
I distinctly remember our first trip without a stroller and how freeing it felt! We would pack a small backpack with supplies for the day. Backpacks can make your back hot, so our solution was to take turns carrying it. I have also packed a backpack for a park day while at home so I could see how much I thought I needed to carry. Without the pressure of getting to the park, I was able to greatly reduce how much was in our park backpack. I'm not sure how old your kids are, but I made sure our pack was light enough for the kids to have a turn carrying it!
There are not places to leave your bag while on a ride, but there are lockers at the front of the parks.
Locker Rental is $9 a day for a large locker and $7 a day for a smaller one. On one winter trip where we knew we would need warmer clothes in the morning and evening, we rented a locker and kept our jackets and ponchos there for the day.
Thank you for stopping by the Disney Parks Moms Panel!
Wishing you a magical day,
Amy H.