Could I 3D model and print a working airbrush in PLA?
The problem with this I see is that the PLA takes and holds paint super well. I have painted it with acrylic a lot and it works great for models you want painted after printing.
I know next to nothing about airbrushing, but it seems to me like keeping a printed airbrush clean for re-use would be a big pain. It looks like they make the pro airbrushes with some kind of stainless steel most likely for the non-sticking purposes of re-use, which PLA just won't have.
That being said, don't be afraid to model it and try! I would make a suggestion that you switch to ABS and then do a vapor bath on the result to smooth out the material and possibly make it easier to clean excess paint out of.