To get the mails your users must have a server connection.
If your users want to download mails, they have to be able to either connect to the server online or, if not possible, connect to a server that has the mails offline.
The only way I can think of is a mail collector service. These usually connect to the users' mailbox with their credentials (which need to be provided) to download the mail. Then you can write rules for individual users.
Think of "somehow forwarding it to the correct email" rule—that must be clearly defined and can only be done with a set of rules saying "mail to userA > forward to userA".
An automated way is not possible, not that I know of.
There is, for completeness, the method of a "mail relay". It takes mail via SMTP and simply forwards them via SMTP to another server, inside your network. This does not work with POP or IMAP as they are reflecting the other mail direction (SMTP = mail from client to server, POP/IMAP = mail from server to client).
This is a) provided that a server SMTP connects to your relay. In case of public mail providers this will not be a solution as the servers will simply not forward mails, gmail, tiscali as examples.
b) your mail relay is receiving mails for a managed mail domain. Then an outside server word SMTP forward mails to your relay, in the DMZ and it could forward to the server in LAN.
c) anything else and bad configuration with a relay is very likely used for mail spamming.