RayLivingston:
the "server" is using one of these, and no AVR Arduino.
Those are great, you can just upload sketches over USB without having to deal with the hassle of connecting a separate USB-serial adapter like with the other modules and you have all the pins broken out and a 4 MB flash chip. I've mainly used the WeMos D1 Mini boards because they're smaller but I had one of those "NodeMCU" boards first.
RayLivingston:
So, it appears I can implement the entire functionality using the ESP8266 library. Though it's not obvious to me how to do UDP on this, as there seems to be no UDP class as part of the ESP8266 library.
Here's the list of the UDP functions:
https://arduino-esp8266.readthedocs.io/en/latest/esp8266wifi/udp-class.html
and some examples:
https://arduino-esp8266.readthedocs.io/en/latest/esp8266wifi/udp-examples.html
And ESP8266WiFi library documentation:
https://arduino-esp8266.readthedocs.io/en/latest/esp8266wifi/readme.html