Currently the only way I know of getting HBO GO on webOS is via the Amazon Prime app:
https://help.hbogo.com/hc/en-us/articles/204872107-Watching-HBO-on-Amazon-Prime
I am not sure whether you have to have an actual Amazon Prime Video account as well, or if you can just install the app and use the HBO GO part of it.
Sadly so many sites and services rely on Adobe's Flash, which is a constantly changing environment, instead of moving to newer more stable platform agnostic technologies. Many manufacturers just do not want to have to keep playing catchup with Adobe.