In the 19th century only non-British people needed to be naturalised when immigrating to Australia.
To be able to vote non-British residents needed to be naturalised. Technically they also needed to be nauturalised to buy land, but in reality land was often sold privately, for example from one German family to another. Often those with non-British backgrounds did not bother to become naturalised for many years, if at all.
When Britain (and therefore Australia) went to war with Germany in 1914, many people born in countries outside of the British empire felt they needed to become Australian citizens, even if they had spent their whole lives in Australia.
The State Library does not hold naturalisation records. These are held by the National Archives of Australia and can be searched on their website.
History of Naturalisation laws in South Australia
Naturalisation in South Australia before 1848 was granted under specific Acts which named the individual applicant.
After that date legislation was passed which allowed naturalisation without a specific Act. Each state retained it's own naturalisation legislation until 1903 when the Commonwealth Government standardised naturalisation laws.
In South Australia before the 26 January 1949 naturalisation gave citizenship to Australia, but with the rights and privileges of a native-born British subject. After 26 January 1949 legislation changed so naturalised Australians became an Australian citizen with all the rights and privileges of a native-born Australian.
From the National Archives website scroll down to 'Search the Collection' and click the link
Click 'Record Search'
Click 'Name Search'
Enter the family name.
From 'Category of Records' use the drop down arrow to select 'Immigration and naturalisation records'.
A list will indicate what is held, and what has been digitised for viewing online.
Copies of naturalisation records can be obtained through the National Archives Adelaide Office.