The approach I suggested does not require the support agents working outside e-mail. For example, you as an admin, configure HelpDesk site in SharePoint and set up all the required triggers. The agents will be notified to their e-mails about new tickets. To add a new comment to the ticket, any of them can just reply to that notification being in their mailbox interface. HelpDesk will process that reply as a new comment in the ticket. If you set up the trigger for automatic assigning tickets to the first agent who replied, his contact will be set as an assignee automatically. He needs do nothing. As an assignee, he will receive e-mail notifications about further replies in the ticket. To add his own, he needs just to reply to the notifications he receives - all of them will be automatically processed as new comments in the tickets.
All replies, both from agents and requesters, are sent to the shared mailbox and then forwarded to HelpDesk. If you set up the forwarding rule to keep the forwarded messages in the mailbox, they all will be available in it. Each message will have the full history of the communication quoted below the current message. Also, it will be available in the HelpDesk site in tickets (which is more convenient to track as for me).
The result: agents just write e-mails, requesters just write e-mails, you see the communication on HelpDesk site.
Perhaps, I do not understand your use case and requirements fully. If so, I am sorry and hope on your patience in explaining it to me.