Hello! I hope that this is the right place for feature requests related to the Android Webview.

I am the founder of Pushpad, a web push notification service. We receive many requests from developers that would like to use the (Web) Push API also for hybrid mobile apps. However the Push API is not yet supported by WebView.

Could you consider to add support for the Push API?
https://www.w3.org/TR/push-api/