http://www.wdwinfo.com/tips_for_touring/coolers.htm
Cooler Policy for Theme Parks and Water Parks.
Magic Kingdom
Small coolers (that one person can carry) that will fit into the lockers under the Train Station are allowed. Coolers will be inspected for glass and alcoholic beverages which are not permitted.
Disneys Animal Kingdom
Allowed ONLY for medical purposes. If a guest is in need of medication, they may bring a cooler and store it in a locker. If it will not fit into the locker, the cooler will be kept at Guest Relations. Coolers are not allowed in the Theme Park. Coolers will be inspected for glass and alcoholic beverages which are not permitted.
Disney-MGM Studios
Small coolers that will fit into the lockers are allowed. Coolers will be inspected for glass and alcoholic beverages which are not permitted.
Epcot
Small coolers (that one person can carry) that will fit into the lockers are allowed. Coolers will be inspected for glass and alcoholic beverages which are not permitted
Water Parks(Blizzard Beach and Typhoon Lagoon)
Coolers of a size that may be safely carried are permitted
No glass containers or alcoholic beverages allowed
No storage areas for coolers