I know that if I have the mac address I can find the vendor of the card but is it possible to find the exact model?
Not easily to the exact model number, but often to a vendor.
Check out this page from IEEE. http://standards.ieee.org/regauth/oui/index.shtml
You want to look up an OUI record. (mac address)
Another hint: I have found that many of my servers shipped at the same time have MAC addresses in the same relative range, so if you have another server on your network with a similar MAC and you know its Model number, you might be able to make a match.
Ah - no. Vendors are supposed to "purchase" MAC addresses, but to my knowledge they can do with them what they want. Which means you can not go back to the type.