OP, I answered in the Swan & Dolphin app, where you also posted your question:
You order a Minnie Van through the Lyft app when you want it. You do not order in advance.
Minnie Van drops you off in one of the bus slots at the Magic Kingdom.
To give you an idea of time: when I used a Minnie Van in October, I was staying at the BoardWalk. It took 15 minutes to get from the BoardWalk to the Magic Kingdom at 8:00 am.
Minnie Vans start operating at 6:30 am. However, my driver told me that some cast members have a shift that starts at 6:00 am and their Minnie Vans start operating at 6:30. Other cast members have a shift that starts at 6:30 am and their Minnie Vans start operating at 7:00. So every single Minnie Van is not on the road at 6:30.