Q: Need to Find what my CD key for Call of Duty 4

Mac App Store apps do not have passwords, serial numbers or alphanumeric keys. If you had a previous version or a trial version of the game installed on your Mac and then downloaded the version from the MAS, you need to delete everything using something like AppCleaner and start over. After everything of the game is gone, then redownload the version from the MAS.

Apple says no.

Do apps from the Mac App Store require activation keys, serial numbers, or registration numbers?

No, but you may be asked for your Apple ID and password the first time you use an app.

http://support.apple.com/kb/HT4461

You should inquire with the developer's support if there is perhaps an in-app purchase that has to be made to make the game multi-player.

Well, then you will need to check with the developer's support as to where you get it, because as you can see from the MAS FAQ, Apple is unaware of any MAS apps needing anything of the sort.