Disney World is private property, just as your home or non-regulated business are, and can ban gum chewing as part of the agreement to enter the property, if they so desire. The CC laws detail which Public properties, or alcohol serving establishments if stipulated, are not permitted to allow CC. They can single out establishments that serve alcohol because it's sale is regulated by state and federal statutes.
My state is now considering CCW's inside of alcohol serving establishments, as long as the permit holder does not consume alcohol. It could get very interesting in Ohio, soon.