I arrive at LAX Tom Bradley International Terminal on Air New Zealand and depart from Terminal 5 on an American Airlines domestic flight. I do not have any checked luggage. Can I transfer from TBIT to T5 without having to go landside? Would I still need to go through TSA in T5?
I'm not sure what you mean by "landside", but if you mean going through immigration and customs, everyone arriving in the US at any airport, even for international-to-international transit, must immediately go through immigration and customs (unless they already went through US immigration and customs at a preclearance airport abroad). There is no departure waiting area for people who have not gone through immigration in the US.
After going through immigration and customs, at every airport, you have to go through a security checkpoint if you are connecting to another flight (domestic or international), but depending on the airport, there is often a dedicated security checkpoint for people connecting from international flights, which does not have long lines like the checkpoints for people coming in from outside the airport.
There are post-security corridors connecting TBIT with terminals 4, 5, 6, 7, and 8 at LAX, so if you go through security at TBIT, you can get to your gate in Terminal 5 without going through security again.