I want to have an additional 120v fan come on when the AC unit is on. Am I able to wire that off the thermostat somehow?
Here is an external, low-tech solution. Put a sail switch in front of the ac unit. Wire the switch in series with the secondary fan. A small open-ended box could enclose the switch and terminals. The airflow would turn it on.
If it is a central ac system you could put a thermodisc at one of the registers or at the plenum. Use a "for heating" model that switches on when it detects cold air.
Disclaimer: I am not an AC specialist so, take this with grain (or sack) of salt.
It all depends on what the function of this additional fan is and how old your system is (things might get somewhat complicated if you have heat pump system).
If it is to supplement indoor air circulation then you should be able to control it from thermostat. Usually thermostat connects red (~24V) wire to green (fan) wire when the fan is switched on. Or, if you want additional fan running only with AC then use red + yellow (cooling) pair instead.
Now, if you have black (common) wire is available, measure voltage between it and green/yellow wires when fan comes on. If you see 24V - great, you are half way there. You can check this question for one simple solution to your problem... or you can continue reading on my somewhat more cumbersome solution.
If there is no black wire, uh-oh... you are in trouble. The problem is that in most setups black wire is missing. What you need then is a way to detect when red wire is connected to green/yellow wires by relay in thermostat.
I think in both cases your best bet is cheap relay module like this one. There are tons of them on internet. Note two things though - there must be optocoupler on board (look for "isolation" in the description and that big black chip on photo), and the relay must be rated for your fan's voltage and current. The coil voltage does not matter, you won't be powering it from thermostat. They are available in 3, 5, 12 and 24V, so just pick the one you can get suitable wall adapter for.
Now, here is where things get tricky. I cannot help you with schematics without knowing the specific board. You have to trace the PCB to figure it out by yourself. But the gist of it goes like this:
if you found black wire (see above) then voltage between it and green/yellow will be your triggering voltage - connect one to ground, another to input and you are done. Otherwise you have to use red and green/yellow pair, and your trigger would be an absence of voltage when thermostat shorts these together. Also your fan will be connected to NC (normally closed) output terminals. Make sure your relay is not overheating, since it will be ON all the time the fun is not working.
In both cases you need about 2.4K resistor in series with either of wires. I would also recommend a diode in series, just in case the LED in optocoupler cannot handle reverse 24V. Any cheap diode with reverse voltage > 30V and current > 100mA would do.
Now, if what you want is to supplement the air flow over outdoor AC unit that is completely different story. Those usually have two pairs of wires coming in - one is mains power and another is controlling voltage, most likely an extension of yellow+black wires from thermostat. So, theoretically you can use those just as above, but you better ask a specialist about it.
Also, if they are indeed black/yellow pair, that would be a nice alternative spot to connect indoor fan to. Good luck and work safe.