Delivery (childbirth) at home, at a home, in a home?

There's a couple of problems with "in a home".

"in" means "inside" in this context, whereas "at" means "at the location of" - it's a bit more general. The delivery will be to the door of the house, it won't be inside the house, but it will be at the same location. So, "at" is better than "in".

The next problem is "at a home".

"A home" is too vague - it makes one think of a specific place, eg someone's actual house, but doesn't suggest that it will be the customer's home. Saying "your home" or "the customer's home" fixes this problem by specifying whose home the delivery will be made to.

You can alternately just say "at home", which is also general but is a familiar phrase which people would take to mean "at their own home".

"No matter if your delivery takes place at home or in the hospital ..." seems fine to me.

"at home" means one's own home. If, for example, you were visiting your parents and gave birth in their home, you didn't give birth "at home" but rather "in a home."

And if you are debating now between "in a home" vs. "at a home" because of the circumstance in which you gave birth outside but still on the premises, just know that "at a home," while not ungrammatical, is very bizarre and unnatural phrasing. It's a construct someone not fluent in English would use. When we hear "at a home" we expect there to be more to the phrase: "at a home for the deaf," "at a home my mother was renting out." A commenter above, BoldBen, is spot-on when he points out: 'At a home' implies somewhere other than the patient's own home and suggests an institution of some kind, either a care home, residential home or, if there are any left, maternity home.

The phrasing of "in a home" is much, much more natural.