I was told that all the oil onsite (including DTD restaurants) is some kind of vegetable oil. Disney won't let them use lard at Totilla Joe's - they have to use the vegetable oil like everyone else on disney property.
The DLH TV channel shows how they then re-use the frying oil to power the vehicles and trains, etc. on property.
Sorry, don't know which vegetable oil is used but it's plant based oil, not animal fat if that's what you want to know.