I lost my Windows 7 product key, do I need to buy a new one?

I upgraded my OS through the internet download option and had a backup disc mailed to me. I recently attempted a clean install as a means to clean out my hard drive, and now I’m being prompted to
insert my product key. Unfortunately I cannot locate the one given to me initially, and it is not included anywhere on the packaging for the backup disc. I have no access to my computer as I must enter my product key before going any further. The Microsoft board is slow to answer my query, and I was wondering if
any sort of workaround were possible. Thanks