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Hi HerschelJ, 

the first thing that came to my mind when I read your post was "SharePoint Auditing".

Now this has been completely changed between SharePoint on premises and SharePoint Online.

Today, Auditing ist still taking place under the name of "Microsoft 365 unified audit log". For SharePoint this means that less data is being written to the audit log, but maybe what you need is still there. See https://docs.microsoft.com/en-us/microsoft-365/compliance/search-the-audit-log-in-security-and-compliance?view=o365-worldwide#audited-activities

In case the opening of reports is written to this unified audit log, you might want to create some sort of web service that runs under an identity that has permissions to view the unified audit logs. And this web service could be called by a simple SPFx Web Part that displays the result.

If the events you are interested in are not written to the unified audit log, this is going to be tough. Then you would have to create your own mechanism for launching reports as you wrote in your post and make sure that this is the only way for users to launch reports. The display of the results would be similar as above.

Just my 2 cents.

Matthias