If they installed Windows on that computer it should have come with a COA. Since it didn't one can only assume that it was an illegal copy of Windows. Every legal copy of Windows has a product key. That is what you are buying.
The Windows install on your computer has a product key. There are ways to find out what it is but I probably can't talk about it any further than that. The problem is would that product key work with the Windows Xp disc they sent you. If you tested it and it didn't work you would be left with a computer without a working copy of Windows.