Is there a way to permanently delete Exchange/Outlook email while away?

Short answer: Nope -- In Outlook/Exchange land, permanent deletes are client-side only.

Possible workaround:

If you want to do it without leaving you Outlook open, you'd probably need to set up a regular (server-side) Delete rule in Outlook, and then write a script (say with Powershell) to empty your Deleted Items folder.

This script would need to be scheduled to occasionally run on the Exchange server itself (or from a workstation that's on while you're away and has remote management access to the Exchange server).

You could set a retention policy that automatically deletes messages in the Deleted Items folder or an alternate folder if you move the messages there.

I am using outlook 2016 and came here to find that it is not possible; however I then continued to proceed with my rule to send certain items to the deleted items and on 2016 there is now an option to then permanently delete items after sending to deleted items.

Step 1 - select conditions

Step 1 - select actions - when selecting to move to deleted items folder the option became available to tick for permanently delete.