Generally speaking, no. Airag is a seasonal, niche, perishable product, that as far as I can tell is not commercially produced and is hence not regularly available at supermarkets, liquor stores or bars.
Visiting Ulaanbaatar just before the Naadam festival a few years ago, I did see a few pop-up stalls selling the stuff, but generally it’s made at home and only sold where produced.
In the picture above, the airag was produced in the ger (yurt) it was sold in, more specifically in the blue plastic barrel in the background. Эрүүл мэндийн төлөө!