Tracking my website visitors

So, my girlfriend has website through Godaddy.com.

I'm assuming that means that Godaddy not only provides the domain name but also provides the hosting - i.e., the Web server runs on a server machine or machines at a Godaddy site. If so, then:

Is there a way to use Wireshark or anything else to find what the MAC address is or even what the IP address is?

...at least for Wireshark or other packet sniffers, the answer is "not from your machine or hers" - the traffic is between the harasser and Godaddy, and you don't have access to it from your machines.

You should ask Godaddy whether they provide any form of logging of the IP addresses from which contact messages are posted. (The MAC address is unavailable; it's only available to the machine to which the harasser's computer immediately sends the packet, which is probably a cable modem or DSL modem for a home user and a corporate router for a corporate user.) Given the IP address, it should be possible to find out to what organization that IP address belongs, and given the IP address and the time the contact message was posted, that organization may be able to say to which user that IP address was assigned at that time.