1.
Set up email server on the computer with Internet access. Each manager will run an email client like outlook, and connect to this server. The server relays the email through the old server you've been using all this time.
I suggest the free Mercury email server. It runs on Windows, and is free with no limits. Relatively simple to setup.
The problem I see here is, if this "computer with Internet access" is not connected to the other computers in the LAN. (Because, if they're on the same network, how come the other computers don't have Internet access?). The simplest solution is to add another network card to this computer, so it has two network cables attached. One to the Internet, another to the intranet.
2.
Connect all the computers to the Internet. But add a firewall. The firewall only allows traffic to the company's email server. Everything else is blocked.
3.
If the manager's computers don't have Internet access, because of security reason (because you're dealing with bank data), then there is no solution. This is by definition. Because any solution will involve some sort of connection, and this will violate the security policy in place.
If the manager's computers are disconnected because of the security policy, then (2) is VERY risky. Yes you will need a firewall. You will also need someone with enough skills to administer it. If you're here asking for advice, than you're not the person, and neither am I
. (1) is also a bit risky, especially if this computer is running Windows. They're notoriously insecure, succeptible to all sorts of virus/malware/hacking. If this machine is compromised, and it is connected to both the Internet and the LAN, then the attackers have access to your Intranet.