Absolutely; as they build out their infrastructure they start doing their own deliveries. We have a second home in Charleston, SC and that area was one of the first for Amazon delivery.Must vary by region, almost all Amazon-shipped deliveries come USPS here. I don't know how much of the USPS infrastructure they use, I've heard they drop ship a lot to the local PO.
The local paper did a write up on how Amazon was establishing their new package delivery system in Charleston and the new facility was right in the back yard of the Mercedes assembly plant for the vans they are using. We had Amazon delivery there for at least a year before it came to Virginia.