So it turns out that you can't add AU's via Synchrony for the Amazon Store Card.
You can, however, give the card number to another user and they can enter it in the "Add Credit or Debit card" interface.
The source of confusion on my part was that the "Add Credit or Debit Card" interface asks for an expiration date which the Amazon Store card does not have. The key part that I was missing is that when you enter a valid store card number the form detects it and removes the expiration date box allowing you to successfully add the card.
So in a credit sense you can't add AU's (as in get it to report on someone elses credit report), but you can let others user your store card which is what I was after.
