Older triplex built in the 1960's, sub panel has no ground bar. Would like to install a dedicated 20 amp breaker and run new wires, what happens to the ground wire if there is no ground bar in the panel to connect too? Would it be OK to leave ground wire caped and just have the neutral and hot connected only? This is how it is done with rest of the outlets, no ground wire.

Would it be ok to leave ground wire caped and just have the neutral and hot connected only? This is how it is done with rest of the outlets, no ground wire. NO! The existing receptacles are 50 years old. Codes have changed just a bit since they were installed.

Is this a main breaker panel, or is there a main disconnect ahead of it?

This is a sub panel. The main shut-off is outside. I like to install a 20amp and run Romex 12/2 wire to hard wire to an AC, what do I do with the ground wire since there is only a neutral bar. Since today code require a ground, does that mean I have to replace the breaker box with a new one? Any other options?

There should be a ground bar in that panel. The only way it is possibly legal is if everything is in conduit and the conduits are the grounds for everything.

IF this is the case you can add a ground bar right to the back of the panel. If you do, make sure it is the same brand as the panel and you use machine thread screws to install it.

I have attached a picture of the sub panel. There is only the neutral bar on the right hand side. I do not see a ground bar.