This is the most common method. Search your email inbox for the following terms. Make sure to check "Spam" and "Trash" folders if you don't see it in your main inbox.
TechSmith provides a License Key Finder tool that can help you locate your Snagit license key: how to find my snagit license key exclusive
Check C:\ProgramData\TechSmith\Snagit [version]\Reginfo.ini for registration details. This is the most common method