I've read the forum, and I know that trivially, the answer to this is "the ones that have METARs reported through aviationweather.gov API." Handily, aviationweather.gov's APIs seem to be used by SkyVector, so if the airport has a colored ball on SkyVector, you should drill a hole for an LED.

Some web searching and an email to the NWS further tells me that if the weather station is funded by the FAA, it includes the network connection from the airport to the system that ultimately publishes it to the API, but if the airport pays for the weather station out of its own funding, it's up to the airport to publish the data to NWS/FAA itself.

Since building my map, I've noticed a few things:

• Small towered airports (RHV and PAO near San Jose) will often only have a METAR when the tower is open, so the LED goes dark at night.
• A couple of airports that have automated weather reporting (E16 and PVF) did not have METARs published through aviationweather.gov when I was drilling holes.  But they do now.  (Fortunately, in both cases, there's a spare LED that's in the right place for me to drill a hole and put through the front of the board.)
• Weirdly, one airport that almost always had a METAR published (C83) hasn't had it's LED lit in something like a week.

Is there a way to get historical METARs and see what airports have reported in the past?  I know on aviationweather.gov you can get a 5 day METAR history, but I wish I could tell which airports commonly report data.