I believe that Chrysler bought the Airtemp Company in the 1930's. Not sure when A/C made its way into their cars, though.
As an aside, back in the day, Chrysler was somewhat unique among the Big Three US automakers in that it tended to introduce new technologies first in its lowest priced lines - at least those technologies that didn't require a lot of horsepower to run. So even the lowly Plymouths got options like electric powered automatic chokes, "floating power", speedometers that changed color according to speed, etc., before the bigger marques got their versions. Obviously however A/C and automatic transmissions were power hogs and I think they probably showed up in Chryslers and DeSotos first.
I think Chrysler did it this way as a test method. If it bombed in the low priced line, no big deal, PR-wise. Once it was proven in a Plymouth or a Dodge, it then was allowed to be put into the TOL models.
All this started to change in the 50's and 60's, when styling became as important, if not more so, than engineering prowess.