The question is how the reports get printed. If you export the report to Word via MailMerge, you could find yourself with no way to handle the required event.
In summary, the issue is that Access doesn't do ANYTHING unless there is an event somewhere to trigger an action. BUT if Access doesn't SEE the event, you still do not get the action to occur. For instance, when printing a MailMerge from Word...
If someone prints an Access report (i.e. a real report residing on the Reports-tab pane), you don't have an event that actually says the report got printed. You DO, however, have an event that says the report is about to be printed. The OnPrint event triggers after the time a section gets formatted for printing but before the affected section is actually printed. This is as close as you can get to the "right" event.
What I would do is put an OnPrint event in the report footer (i.e. a footer that will be printed on the last page of the overall report.) Not a page footer, not a detail section, and not any group header or footer. Only the report footer is even close to being guaranteed as being on the last page of the report.
So in this event routine, open a recordset, do an .AddNew to create your log entry, write the fields you want written, like report name and customer name and date. Do the .Update event. Close the recordset. Return from the event routine normally (i.e. do not attempt to abort printing.)