• For incoming mail, on your server run a mail retrieval agent like fetchmail to fetch mail from the externally hosted mailbox into a maildir on your server.
• To serve that maildir to your clients, on your server run a mail delivery agent like the IMAP server Dovecot.
• To accept outgoing mail from your clients, on your server run something like Postfix with a relayhost configured with the details of your externally hosted SMTP server.

There’s nothing unusual or tricky about any of this arrangement.

(Please don’t do this. Pour the old oil into the empty container and return it to the store for responsible disposal, or municipal hazardous waste facility, or whatever scheme exists in your location)