Based on my own experiences with this sort of thing...

When connected through your VPN, your company email server (mail.mycompany.com) knows you are connecting from the "inside". It's happy to send mail to both internal and external addresses. The technical term for sending mail to an external address is "to relay mail".

When you're connected to the great Interweb, you're outside your company network. As such, your company email server is not very keen to just relay stuff for you, unless you can identify yourself well enough for it to be happy that you are you. From outside, your email server is happy to send mail to intenal addresses, just like other mail connections from anywhere.

You'll need to enable things like SSL/TLS for your outgoing mail, and have some form of authentication.

On Outlook 2003, go to "Tools -> Email accounts -> View/Next -> Change -> More Settings -> Outgoing Server (tab)"

Here's there's a checkbox for "My outgoing email server requires secure authentication". Give this a go.

The SSL/TLS settings are on the "Advanced (tab)".

I spend my life connected through the office VPN, even when I'm in the office. This way it's always the same.