You don't say where the family was located but here in the UK only live births got birth & death certificates
If the baby was stillborn, it would be on the stillborn register -- which is not open to view. Only specific relatives can apply for a copy of a stillbirth certificate:
http://www.genuki.org.uk/big/eng/LIN/civilreg.html#Stillbirth :
Stillborn children were not registered prior to 1927.
Stillbirth registration was introduced on 1 July 1927 to help protect infant life, provide a valuable source of statistical information and to give parents the opportunity to have their child officially acknowledged. A stillborn child is a child born after the 24th week of pregnancy who did not breathe or show any other signs of life. When a child is stillborn the midwife or doctor will issue a medical certificate of stillbirth which will be used to register the stillbirth.
When stillbirth registration was introduced the the age limit was the end of the 28th week of pregnancy, not the 24th (as it is now). This is a relatively recent change following the greatly increased survival rates of premature babies.
Current GRO policy on obtaining stillbirth certificates: "Due to the sensitive nature of stillbirth registrations, the procedure for ordering a certificate of the entry differs from other types of certificates. We will only send out the application form after we have been contacted by phone or in writing by the mother or father (if he is named on the certificate). In cases where the parents are deceased, a brother or sister can apply if they can provide their parents' dates of death."
OOOPS just realised you have posted this on the Australia board, so guess it is Aussie records you are looking for -- sorry!