Thanks to iStats Menus I noticed there was a process constantly using just enough CPU to attract attention. Activity Monitor showed that one of the VS Code helpers was hovering at around 7%. Not terrible, but it was annoying me.
After a little trial and error, disabling and reenabling extensions, I discovered it was the Settings Sync extension. Once that was disabled things calmed down. I dug a little deeper and discovered that it was turning on the
sync.autoUpload setting (which is off by default) that was really doing it. With that set back to
false the extension ran fine without using a bit too much CPU for my taste.