<p>I could have sworn I read somewhere that when printing with TPU to make sure the part cooling fan is blowing. But I just did a quick Googling and couldn't find anything stating such on Matter Hackers or All3dp.</p>

<p>I currently don't have a part cooling fan attached (waiting for square nuts to come in). I've been able to get by printing PLA without the fan. I'm curious if this is going to be a major obstacle with TPU.</p>

You'll probably be fine printing TPU with no fan. I just started printing with TPU, and did a lot of test prints to find out what settings work. Fan made little difference. With hotend at 230 °C, which I started out with, 0-20% fan was fine. I eventually increased temperature to 250 °C, which made extrusion more consistent and allowed me to reduce linear advance K-factor somewhat, and at that temperature having a bit more fan (I'm using 40% now) seems to help the material hold its shape, but it mainly made a difference at higher print speeds (over 35 mm/s) where the motion of the nozzle was "pulling on" the still-very-soft material just extruded. At 30 mm/s and below, fan still doesn't seem very important.
All of this is likely to vary somewhat with the properties of your machine. However I think it's safe to say you should be able to find a combination of print speed and temperature that make it possible to get by with no fan.
Follow-up:
Upon further experimentation with TPU, I would say you really
don't want any fan at all
, except possibly for bridges. I've found significant distortion to shape just from air pressure from the fan, and at higher speeds the fan makes the print brittle just like what happens with PETG. Layers of TPU really seem to want time to melt together to bond, and without a fan blowing on them they don't seem to lose their shape during that time.