I use Amazon a lot and have never seen a private vehicle deliver anything.
The private deliveries are a new thing with Amazon, and they seem to require only a clean drivers license and an insured vehicle. They deliver this way all over the place in Jacksonville now.
I have a co-worker who got an Amazon delivery from a private vehicle. There was some kind of marking added to the vehicle.
He first thought that the driver sitting in his driveway was lost and checking location but it turns out his son had ordered something.