Disney refers to the people visiting its properties as guests. The idea behind this name is that visitors are not simply customers engaging in a transaction with Disney but are in fact each honored guests.
You may also hear the specific phrase “guest of Mickey” when referencing children ages two and under. Children this age are able to enter the theme parks free of charge with a paying adult. They are also able to eat at buffets for free, again, with a paying adult. In this sense, their visit is considered “on the mouse.”