Is there any option to log users who done playback export and create a report of it.

Yes.

This is the audit log, you find it in the Management Client.

As you are asking the Developer Forum I assume you want to do this in code and fortunately there is a small sample you explore on how to read the audit log.

https://developer.milestonesys.com/s/article/read-Audit-System-or-Rule-logs-from-the-MIP-SDK