After several years of use with no problem, the OpenWeatherMap entry in the Admin Console disappeared and I can't figure out how to restore it. Any suggestions?

@JMC were you by chance using PGC? That was shut off earlier this year please see this post. 
 

If you have Polisy or Polygloy availability you can run a local version if available in the node server store. 

 

I don't use either Polisy or Polyglot. Historically, OpenWeatherMap was prominently listed in the ISY Admin Console but it disappeared a few days ago. I would like to get it back but I haven't been able to do it yet.

Sounds like you were using Polyglot cloud to get OpenWeatherMap data.  As @Geddy mentioned it was shut down 6 months ago due to security vulnerabilities.  Also Polisy's also became unavailable earlier this summer due to the global chip shortage.  UDI is working on a replacement that's due to be available in the coming weeks.    There were at one time some heavy hardware discounts for those that were using Polyglot cloud, you'll have to check with sales@universal-devices.com to find out what's available now.