What should a young man do who wants to marry a girl, but she and her family object to him keeping his beard, while he wishes to keep his beard and marry her at the same time?

Shaving the beard is forbidden (haram), and it is not permissible for you to shave your beard for this purpose. Either the girl's family accepts you with a beard, or you forsake marriage to their daughter. And whoever gives up something for the sake of Allah, He will compensate him with something better.