Please tell me, could a doctor bring home vaccinations to give to his family?

Depends on variables: Some do, especially with such vaccines as the flu. However, it is considered inappropriate unless the doc maintains a proper medical record of all such events. A record of the time, date lot number of vaccine as well as where it was given and some vital signs and or other information. Failure to keep track of such treatments can subject the doc to sanctions by their state's medical board.