Android comes with two webviews (SetWebViewClient and SetWebChromeClient) and it's painfully obvious that the regular web client has zero websocket support.
However, I'm not sure of the capabilities of the WebChromeClient, except that Chrome on an Android browser seems to support Websockets.
Can someone confirm if WebChromeClient supports Websockets?
(This particular answer didn't provide enough info about WebChromeClient What's the difference between setWebViewClient vs. setWebChromeClient?)
There is a simple test to check if your browser has websocket support. Launch below url and ping the public hosted service. If you are able to ping from your browser it has support :)
https://www.websocket.org/echo.html
I have put these details on my blog, here.