Is it possible to send push notifications to Android without building any application?

The answer to the question is simply NO, you need to register an app to google C2DM server, to be able to send push notification to this device.

It depends on your requirements. You can use third-party service like Pushed (https://pushed.co/) or Spontit (https://spontit.com, https://api.spontit.com).

Or you can setup the website as a Chrome-app on the phone and then you'll be able to send notifications through Chrome.

https://developers.google.com/cloud-messaging/chrome/client https://developers.google.com/web/updates/2015/03/push-notifications-on-the-open-web?hl=en

In both scenarious you'll need to advise your users to install the app on the phone which will subscribe to notifications.