Thanks for all the help so far.
The firewall should be turned completely off.
I was able to install VNC and connect to the computer with that. In a few hours, when the employee will be done with the computer, I will VNC into it again and poke around the settings. I will check the netstat -a command, as ajbritton suggested.
Mattx, this computer is part of a domain. I will see about cleaning up profiles and other random junk on the computer and see if it helps.
I will post back later with an update.
Thanks for the help!