Considering it's not compulsory to get immunised, and you have approximately 300,000 births a year in Australia (minus the male births) and they have a child aged between the ages of 2 and 4 to investigate. That's a lot of records to search. But i like your thinking kyja, it might take some time.
A few people have mentioned in earlier posts, they think this child's birth may not of been registered and i can agree that might of happened in some extreme circumstances. but the reason i think there will be a record of this little girl is because of the BABY BONUS.
There is also the law.... When a child is born the parents have 60 days to register the birth before incurring fines of $1000 which then proves a deterrent if forms haven't been lodged. That wouldn't be a deterrent though if it was a home birth and no pre natal care.
A few of you may remember the headlines - "A rise in child abuse because of baby bonus" "Teenagers getting pregnant because of baby bonus" "Babies abandoned because of baby bonus"
The baby bonus scheme introduced by the Federal Government of Australia in the 2002 budget was aimed at offsetting the expenses associated with bearing a child. The scheme was also introduced as a means of increasing Australia's fertility rate and to mitigate the effects of Australia's ageing population.[3] In the 2004 budget the bonus was raised from $3,000 effective 1 July 2004 to $4,000 payable in 2007 but indexed to inflation so that in October 2007, the amount receivable per eligible child was $4,133.[4] The bonus was paid in a lump sum to a nominated financial institution. From January 1, 2009 the payment is paid in 13 fortnightly installments. The receivable amount in January 2012 was $5,437. The receivable amount in September 2012 is $5,000 in 13 fortnightly instalments (parents will receive a higher first instalment of $846.20 and 12 fortnightly instalments of $346.15), or if the baby died or was a stillborn, parents may ask for their Baby Bonus to be paid in a lump sum instead of fortnightly instalments
Hopefully there will be some kind of trail to find this little one, through immunisation records like Kyja mentioned, school records or centrelink paymets