Yes, you are allowed to bring in a small cooler for snacks.
Suitcases, coolers or backpacks LARGER than 24" (61 cm) long x 15" (38 cm) wide x 18" (46 cm) high are NOT allowed in the parks. So, as long as the cooler is "tiny", you should be fine.
FYI, according to Disney's Park FAQ list, items that you may not bring into the theme parks include but are not limited to:
-Items with wheels, such as:
- Wagons
- Skateboards
- Scooters
- Inline skates
- Shoes with built-in wheels
- Strollers larger than 36" x 52" (92 x 132 cm)
- Suitcases, coolers or backpacks with or without wheels larger than 24" (61 cm) long x 15" (38 cm) wide x 18" (46 cm) high. (Coolers required for medication may be stored in a locker or at Guest Relations.)
- Any trailer-like object that is pushed or towed by an ECV wheelchair or stroller
- Alcoholic beverages
- Weapons of any kind
- Folding chairs
- Glass containers (excluding baby food jars and perfume bottles)
- Pets (unless they are service animals)
In Disney's Animal Kingdom theme Park (for the safety of the wildlife), balloons, straws and drink lids are not permitted.
If you have any follow-up questions, please feel free to submit them using the "ask Bret a question" link on this page.