Assuming it's ordered from a "personal" account, then not that I can think of. Amazon stopped putting packing slips in boxes a long time ago, and they won't give out order information. I've tried to find out who placed orders when I received a couple random items, as well as when employees place orders using my work credit card but forget to fill out PO's.
However, on the off chance you have an Amazon "Business" account, then they still put packing slips in the boxes, but looking at the one I just got, that still doesn't have any identifying information, but it does have the Order number on it, but I don't think there's anything you can do with that as the non-account holder.