Is it legal to use a home-made Airband radio receiver in the United States? I work at a local small airport (very small, if we get more than about 10 or 15 airplanes taking off/landing it's a "busy day") and I mow the runways (we have multiple grass ones).

Because of the visual obstructions of the tractor's cabin that I use I have to be even more careful than normal to check for in-coming planes and so I started looking into getting an air band radio for the tractor. However, they can be fairly expensive - not something I'd invest in unless I absolutely had to - and so I started looking into making my own.

I'd figured this was possible because I'd occasionally picked up communications on a cheap radio I had around my house (I live quite close to the airport, about a 5 minute bike ride). So, I went searching, and sure enough - this is a thing that is possible and has been done.

I found this post on Electrical Engineering.SE, Modify an AM/FM radio to an airband receiver, describing basically what could be done; and then this video done by MAKE, Weekend Project: Aircraft Band Receiver, showing it actually being done to cheap transistor radio.

Now, what I'm wondering is, is this a legal thing to do - or does the FAA have regulations on this sort of thing?
FYI, the runway is supposed to be closed XXX with NOTAM when it is mowed.

There is no FAA or FCC limit on constructing homemade airband receivers in the United States.

You can also construct transceivers. As long as you do not sell a transmitter you are allowed to make and operate up to 5 of them.

The exemptions covering low power transmitters are covered in Part 15 of Title 47 in FCC regulations.

The maximum unlicensed radiated output power for airband is 500 μV/m @ 3 m. Typical commercial airband handheld transmitters go up to about 1 Watt and base stations up to 50 Watts. For your purposes a 1 Watt transceiver would probably be reasonable.

Note that typical receiver/scanner designs have an oscillator in their mixer that can generate a small amount of RF noise. Normally the signal is so weak it is completely undetectable. Nevertheless, there are specialized receivers that have no oscillator that some people use to listen to airband with zero chance of interference.