Hi Matt,
I am not sure if I understand your question correctly.
I think perhaps you want to send the voicemail to the email client(eg. Outlook) of the voicemail User and delete it from Unity Connection. Is that Right??
If my understanding is right, then the solution is fairly simple. You can first define a SMTP relay address of the user and select "Relay the Message" option for Voicemail in the Message Actions page for the user.
"Relay the Message" option ensures that when a voicemail arrives for a user, Connection forwards the message to the address that you specify in the Relay Address field.
We can discuss it in a little more detail if you can confirm the problem first.
Regards,
Saurabh Agnihotri