Is a wife entitled to disobey her husband and obey her family by going to her family's home with every birth, and is she sinful for doing so?

A wife leaving her husband's house without his permission, when there is no necessity, is an act of disobedience and insubordination (nushooz) that nullifies her right to financial support (nafaqah). Therefore, it is not permissible for your wife to go to her family's house for childbirth without your permission. However, if she is unable to care for herself after childbirth, you must provide someone to serve her.