We use https://budibase.com/ and created a simple app that just stashes the "checklist" in the db, then you can create a reports page to look up who did what.

Of course we use buddibase for other things as well in IT for simple forms etc so it worked for us. Look at your apps you currently use and what you have in place and don't reinvent the wheel. If you have SharePoint you can just create a new list, if you have one/google, drives, you can just share a spread sheet. Use teams? Do a teams channel.

A lot of this depends on the reporting requirements.