I don't know about birth play, but there is a documentary about orgasmic childbirth. Yup, it exists - I know women who have had ecstatic births, and who have orgasmed while in labour and giving birth.
A lot of what I've read on this message board about birth seems to be based on childbirth as it is in America, which is hugely medicalised. Other countries have different models of care where midwifery care is common and childbirth is seen as a physiological event.
Orgasmic birth is rare, but definitely possible, and is most common in non-medical settings.
There's also such a thing as an epi-no, a blow-up device that women insert into their vaginas and practice pushing out, in order to stretch their vaginal tissues - and yup as the name implies it is intended to prevent episiotomies. I don't know that there's any evidence that it does, but it could be used for birth-play, along with sensual perineal massage. The vagina is an amazing organ, and will stretch considerably!
Also if you find birth sexually arousing then have a read about the ecstatic hormones, a GP obstetrician called Sarah Buckley who had home births has written about it extensively, she has some articles available for free on her website.