I would share a gmail account with them. They could "respond and file" what they are authorized to handle. Then they would "star" anything that is left for you.
That would be using the gmail options to pull in POP mail? (not used them, but I guess that is what you mean)
Going to make some assumptions with that...
- When you hit reply to any e-mail within gmail, regardless of which POP account it came from, the reply from address is the same as the sent to address
- Once an e-mail has been actioned via gmail they can push a button to remove it from the POP server
- I can periodically download to a local e-mail client the inbox, folders and sent folder from gmail for each POP account (in other words have the same local history as if I handled it all with Outlook on my machine)
Have them access the email, use another email account and have your employees send the important stuff your way. I guess EGOL suggested something similar, too 
Sending the stuff to me is good as I don't want to use Gmail as my primary email client.... I just need to remember to edit out any comments my staff my have made before 'fowarding' (cos I can't reply) my reply to the customer.
These suggestions are workable, but just seem a bit cumbersome... but maybe I don't have any other options!