I don't know of any state that allows that.
From an NYC website (http://www.nyc.gov/html/doh/html/vr/vr-online-order.shtml):
A credit card, debit card, or electronic check may be used only to order birth certificates for you and your children, provided you are named on the birth certificates. To request a birth certificate for any other person, including any child where your name does not appear on the birth certificate, you must submit an order in-person or by mail.So it sounds as if they still allow for the possibility of requesting a birth certificate "for any other person." I'm quite sure when I obtained mine (many years ago), I just told them my name and my date of birth (and maybe the hospital too) and sent them a check. Of course my name was on the check. I do remember people talking about how stupid the whole birth-certificate-as-identification thing was because you could just walk into a cemetery and find someone about your own age who had died and request the dead person's certificate.
ML/NJ