We are not aware of any halachic responsa on home birth per se. Childbirth in and of itself is considered risky (See Shevet haLevi 6:128 and, for the opposing view, Igrot Moshe Yoreh Deah II:74.) and thus one is allowed to perform otherwise prohibited actions on Shabbat for a woman in childbirth. The halachic permissibility of home birth would depend on the safety of the procedure relative to hospital birth. Not placing oneself at extra risk is a halachic consideration.

While many physicians stress the potential danger of home birth, it is not clear cut that a home birth attended by a trained midwife in a low risk pregnancy with a clear plan for getting additional help if needed is more dangerous than a hospital birth. A woman considering home birth would need to get professional opinions (both from a physician and a midwife) related to her particular case and then ask a specific halachic question.