I made good experiences with issue trackers. Kind of leading is probably OTRS (https://en.wikipedia.org/wiki/OTRS -- it's open-source despite the horrible website which I won't link here) and RT. However, also software like https://trac.edgewall.org/ and https://www.redmine.org/ provide E-Mail interfaces which work quite well.

The thing with issue trackers is: They are made exactly made for a use case as you describe it. They excel for managing the contact "workflow" with clients, especially if different people work on the same "inbox".

In my experience, Introducing an issue tracker to a team always took some days to weeks of familiarization, but it always paid off.