Technically it is illegal to sell liquor or beer here on Sundays; however, with an added fee, a retailer can get an exemption to the law with their license. So I know how that goes, and besides, like you said: Disney is Disney.
So to sum it up:
Go to an "ABC Liquor" for spirits, during the daylight hours except on Sunday.
Publix or a gas station (I don't understand that--that's illegal here. Doesn't that encourage you to drink and drive?) is the place to get beer and wine, up until 2am.
Don't plan on partying it up at the bar past 2am.