Pulled radio to replace with new radio, but I had to reinstall old due to not having correct connectors. After radio and hvac unit are all back together and battery reconnected, car doesn't start. Lights work. Radio works. Locks work. A jump from another car is a no start. All I hear are a few buzzes and clicks.

Possibly a starter issue?

I guess my other option would be to clutch start it.

 

Have you checked both ends of the battery cables. The ground strap is known to corrode.

 

The odds must still be on a problem associated with your removal/replacement of the radio/climate control units. Before you go too far down the guessing game (and,potentially, causing more damage by jump starting incorrectly?) go back over your connections. The car will not start if the climate control module is not properly connected.

 

as Astromorg says recheck every connection you removed first before looking further.
Were you anywhere near the auto inhibit or inertia switches?
try starting in Neutral if auto.

 

Since this is Jr's car, and he was the one pulling stereo and reinstalling, my bet is he didn't correctly connect one of the cables. He mentioned having to remove the boot around the stick shift to gain access for removing the ash tray, so it could be an issue there. I will have him pull the unit again today (once we buy the correct connectors for the new stereo) and reconnect wires, without buttoning up whole system, so we can assess connection issue.

 

Exactly what joesoap said. I had the exact same issue. Turned out to be the climate control connection what not tight. Recheck all the connections!