The problem with PWA is that when the browser is closed (or inactive for smartphones) the user won’t be able to see a notification.
Optimally the PWA icon should receives a number++ and a ring tone to catch the users attention.
It is possible at all? if not what solutions are there?
What you need is native push notifications.
Vaadin doesn't have built-in support for push notifications, but it is possible to integrate it using the generic JavaScript integration support. You can find an example of this in https://github.com/marcushellberg/vaadin-web-push. You can follow issue #4747 to get updates related to official Vaadin support.
One thing to consider is that this does currently only work for Android devices, but not for iOS. (Update: As of iOS 16, push notifications are also supported on iOS)