Is it legal to use a home-made Airband radio receiver in the United States?
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.