Just set up each user's OE, to access that mail account. For each user's mail account, go to Tools, Account, enquiries account name, Properties button, Advanced tab, check 'Leave a copy of messages on the server'.
This will allow all users to download the same messages from this account. Now you have to decide how you will get the messages off the server. I recommend that you you set all the accounts to remove the messages from the server within some reasonable time frame - 3 to 5 days maybe? This needs to be long enough after the first user accesses the new messages to allow the last user to access these messages before they are removed. If you set this time too long, depending on usage, the mailbox capacity may be exceeded before the messages get removed.