Stock piaware will try to start it if it doesn’t appear to be running, but whether that works depends on how things are set up. Did you install piaware+dump1090 from a sdcard image, or packages, or manual install?
So, how do I now add dump1090 and the netcat command to the startup items? I’m guessing I need a bash script in either init.d or rc.d. Can I use one script to launch both?
That’s the traditional way of doing it, yep. Up to you how you split it up. I just start my tunnels from /etc/rc.local since they’re pretty simple.
dump1090-mutability (here) has package-managed startup scripts & configuration form dump1090 if you want to go in that direction. You’d still need to do the netcat bit yourself.