Hi Jill,
You are absolutely welcome to bring snacks or even lunch into the
Walt Disney World parks. You may not bring alcohol from the outside or glass containers (except for baby food); otherwise, sandwiches and other snacks are permitted. You'll find the
rules and regulations here.
Bringing a meal can be a great way to save money and to accommodate picky eaters or those with special dietary needs. I've seen families bring everything from a loaf of bread and a jar of peanut butter to fully cooked hot meals in small coolers tucked into a stroller basket.
If you won't have a stroller, your idea to use a locker is a good one so that you don't have to carry a heavy bag or cooler.
Lockers are located near the entrance at all four theme parks.
In addition to snacks, I like to bring a refillable water bottle, too, to save money and stay hydrated. You can request free cups of ice water at all quick service counters to refill your bottles.
Thanks for visiting the Disney Parks Moms Panel with your question, Jill. Please let us know if we can help with anything else. Enjoy your vacation with your grandchildren!
Wilma