Camoron,
There are a few ways to determine whether or not the system can access the Internet and as Acejavelin stated you can verify with the NTP service, but there is a way to do a ping from the Mitel 5000 that will tell you whether or not it can resolve the name and actually reach the destination.
If you open up an SSH connection to the system and then go to Command Prompts you can do a ping smtp.gmail.com or
www.google.com and it should come back with an IP Address and whether or not it can ping it.
As for the vmail log that is the wrong log to be looking at for smtp issues. Open up your System Administration and Diagnostics tool and connect to the system. Then press the play button and use the drop down button to open up the tabs. The first tab is Favorites which should have a System Information box within it, if it doesn't you can find that box under the System Information Tab as well. At the bottom of that box you will find a View System Logs button that will show you all of the logs the system is keeping. If you scroll down that list of logs you will find one for NTP and you will also find one for mail.log which will have any SMTP error information that you would need in it.
Hopefully that gets you further along with your testing and figuring out what is going on, but if I had to guess I would say that the system doesn't have access to the Internet or the IP Settings are no longer correct for the new site.
Thanks,
TE