The paid version of Google Apps implements shared mailboxes (poorly) through Google Groups. This would allow you to use a "support" account that would receive emails from outside users, and then display those messages in Google Groups. You can set security for your users to log into the board and reply to the messages as themselves or on behalf of the shared account. Any replies would be logged and so you can see both sides of the conversation. They also support digest emails, so supervisors can have a report of all the day's activity.
You can view more details on Google Apps on their hyper-technical getting started guide, or focus on the shared mailbox feature here