I made a slight mistake - there are actually three programmes.
There's the cultural representative programme - where you spend a year representing the UK by working in the UK pavilion in Epcot. Here you'll either be working in food and beverage, or merchandising (or a third option where you're technically employed by a separate company and work in the historic research centre). Applications open in spring and autumn each year, and if successful you could go out at any time during the year. No need to be a student or anything, except for HRC where you must hold a degree and be interested in history. You just have to be at least 18 and hold a passport for the country you're going to represent. Best to do it post or pre uni - although if pre uni, be aware that it's highly competitive, and most people take at least a year between applying and going out there.
Cultural exchange programme - summer programme between uni (or during the summer immediately after uni iirc) - you could be doing pretty much anything in any of the parks (except face characters as there's too much training needed for this). No specific degree needed, you just have to be a student to be eligible for the visa.
There's also a third one where you can spend your placement year of a hospitality or tourism type degree working out there, where your job would be related to the degree. I know much less about this one though