You have to purchase (and download) the upgrade from the PC that qualifies you for the upgrade. This is the point where it will ask you for your product key.
BUT once you have the download, you can turn it into an ISO and install it on any qualifying machine. (ie any machine running genuine Windows). It is also worth pointing out that an unactivated copy of 7 (ie within the 30 day trial period) is running genuine Windows. OEM or not, it doesn't matter. The win 8 key you get is a retail key, and the previous Win 7 key is still valid as well if you choose to install it on a different machine.
Personally, I didn't even buy mine from the Win 7 laptop I bought, I bought it straight from the machine that was going to Win 8, but it didn't actually ask me to prove I had bought a computer. With the new rules they may enforce that bit.