I am looking for a more comprehensive site for Building Shapefiles for Sydney and its surrounding suburbs which I can upload into QGIS. I have already downloaded the shapefiles from Geofabrik (https://download.geofabrik.de/australia-oceania.html) however there are chunks of areas missing (as seen from my attached image). Any help would be really appreciated.
Unfortunately it is not free and charges are usually based on the size of the area you're seeking. Given the map in your question above, I'm going to guess that it will be prohibitively expensive (but doesn't hurt to ask).
There's always the DIY option if you have some data science skills; Examples.
Microsoft recently released building footprints for all of Australia. See their Github page here; https://github.com/microsoft/AustraliaBuildingFootprints
The licence is Microsoft's Open Database Commons, meaning you can use the data as you need but you must attribute them, see licence conditions here