A while ago I posted a work-around for a problem I was having with a Web AppBuilder application. Working with Esri tech support, we determined what I was doing to cause the problem.

In my config.json, I was using a relative path to the proxy. Note that I had the proxy as a separate application at the root level of our domain because I intended to have a shared proxy for all of our applications instead of individual ones. So the WAB application would be using http://ourarcgisdomain.com/proxy/proxy.ashx.

"httpProxy": {
	"useProxy": true,
	"alwaysUseProxy": true,
    "url": "/proxy/proxy.ashx",
    "rules": []
  }

Well, that was causing the hiccup in the Basemap Gallery Widget. I simply modified the url parameter to have the full path of our domain, undid my custom code, and the Basemap Gallery Widget was happy.

"httpProxy": {
     "useProxy": true,
     "alwaysUseProxy": true,
     "url": "http://ourarcgisdomain.com/proxy/proxy.ashx",
     "rules": []
   }

Score one for Esri Tech Support.

Although, in my defense the documentation on the configuring the proxy is silent on the pathing.