It's not "hard sided" coolers that are prohibited. (They're allowed.)
It's ROLLING coolers (push or pull, on wheels) that are restricted.
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http://www.wdwinfo.com/tips_for_touring/dress-code.htm
Prohibited Items
The following are not allowed to be brought into the Disney Theme Parks:
• Items with wheels, such as wagons, skateboards, scooters, inline skates, shoes with built-in wheels, two-wheeled or three-wheeled conveyances, strollers larger than 36" x 52", suitcases, coolers, or backpacks with or without wheels larger than 24" long x 15" wide x 18" high (coolers required for medication may be stored in a locker or at Guest Relations), and 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. Note: they now have biodegradable paper straws.