There is no need of a key if you purchase Groupy through Steam. You just need to log in into your Steam account to activate it. If your groupy is asking for keys, this usually mean that you have none steam version of Groupy installed. Please uninstall those before installing Steam version. Better yet just purge it all and install a steam version.
Please try purge and reinstall steps as mentioned here: https://forums.stardock.com/488142/groupy-support-faq#reinstalling . Just make sure to follow the Steam part of this guide. Reboot after the purged. Reinstall, and Reboot again after the installation. Retest it. Report back here.
Stardock Community Assistant.