As of iOS 11.3 there is serviceworker feature supported
Even though it has no support for notification. I wonder if it possible to just run any realtime listener, such as firebase database, to receive realtime data and use ServiceWorkerRegistration.showNotification to show local notification instead
Is it possible?
While iOS don't support Web Push notifications yet(as of Apr-2018), all you can do is have an event listener for data from server side(Firebase in your case) and receive that data to show it as an Notification/alert or how ever you want to present inside your application(not in iOS notification area).
Unfortunately, this is the best way you can handle notifications for iOS. I know, its not so great for developers :) Hope Apple speeds up on PWA. They are more than a year late to Service worker party itself. What to expect!
Update : One more link on WhatWebCanDo.today and one from Apple which says iOS-Safari don’t support push notification as a web standard defined by w3c. But looks like there is a OS X safari proprietary solution available.