Hi Carolyn! Great question! You might want to take a look at the
Park Rules and Regulations to ensure your bags are the correct size. Also keep in mind, glass containers are not allowed in the theme parks or water parks. Based on your question asking about a small backpack or cooler, I don't anticipate you having any issues but I want to make sure you have all of the information available.
It is rare that my family travels to a theme park or water park without a backpack. In it, we carry everything from water bottles to sunscreen to rain gear. Our backpack is similar in size to what a student would carry to school. We typically carry it with us and have never had a problem taking it on an attraction. There are a few attractions where it stays on the ground in a safe area but for the most part, our backpack travels with us.
On days we want our backpack in the park but don't feel like carrying it with us, we
rent a locker. The lockers are available on a first-come, first-served basis so I would suggest you get your locker early in the day if you feel like you are going to want one. They are available at the parks and and there are different sizes available. At the theme parks, you can select an all-day-access key-operated locker or a single-use locker that is coin-operated. The key-operated lockers come in small and large sizes and vary in price from $7 - $9 per day + a $5 key deposit. The water parks feature keyless lockers ranging in price from $10-$15 per day. The lockers are the perfect size for storing our gear.
Thanks so much for your question and please let us know if you need any other tips or advice as you plan your magical Walt Disney World Resort vacation!