First of all, I am very happy to hear that your family’s visa got approved and that you finally be reunified soon!
I am afraid, however, that the immigration office (definitely not) or the job center will most likely not support you with the flight costs. Instead, I would recommend you to contact a local refugee counseling office and/or the the refugee council in your federal state. Some organisations have a fund for these purposes and the counseling office could provide you with an individual advice as well as support you with the application.
Let us know if you need help in finding such an office close-by. For this, it would be good if you share your approximate location.
All the best,