2scents: I assume it’s permitted because they asked a Rav, and were told it’s allowed. I have a cousin who is a doula, and I once saw her answer her phone on Shabbos, and then run out to drive to the hospital.
I did hear that a member of the Eidah Chareidis in Israel allows a doula to travel to the hospital on Shabbos.
I did a quick search, and found the following reason (and please note that I am not a posek, this is just a reason I found!): the Gemara says it’s allowed to light a lamp for a woman in labor on Shabbos, even if it’s not medically necessary. The Mishna Berurah says the reason is that a woman in labor needs peace of mind, and if she doesn’t have it, it’s considered pikuach nefesh.
When a woman has hired a doulah, and the doulah isn’t able to attend the delivery, it falls into the category of the peace of mind of the woman. Therefore, as with the lamp, chillul Shabbos is not only allowed, but is required, as in any case of pikuach nefesh.
Of course, you do thigns with a shinuy – like arranging before Shabbos to have a car service on standby. I did read that a doulah told her Rav that because she waited for the cab to come, she missed the delivery, and her patient was extremely upset about it. Her Rav then gave her permission to drive herself.
I know that I was in a shiur when I was in Yeshiva, given by R’ Yaakov Reisman (from Far Rockaway), and he said that if it’s difficult to arrange for a cab when your wife is in her 9th month, there is no problem with driving yourself.