Fast forward to later today, and I got an email headed up "Order XXXX is ready for pickup". It had stopped raining, so I walked over to the Bond and Bond and was told that they wouldn't get the order details through until tomorrow, and since I didn't have a printout of the receipt from online, they couldn't give me the item I'd ordered.
It didn't say anything about needing a print out of the invoice on either of the emails I got, but then it also didn't mention the need for photo ID, which I remember the website flashing a message at me about at one point. Their customer service FAQ only says about the need for photo ID.
So, uh, no headphones.