Thanks to Robert for the reference to their web site!
The Amenities page notes (near the bottom in a small box) that there is a USPS drop between Baggage Claim 1 and 2 at the Information Counter.
There are also FedEx and UPS drops near the Copper Horse Market. (That sounds like a nice market; pick up a few staples post-flight before you head out!)
Stamps can be bought at the nearby gift shop. I’m guessing it’s the Montana Gift Corral (the terminal map is really basic) but I’m not 100% sure.
Last updated 18 August 2018