I'm developing a web-app and I want to send push notifications to users on iPhone. Is there any way of doing this from the browser? Chrome or Safari does not matter.
As of today a web app cannot receive push notifications on any iOS browser (Safari, Chrome for iOS, ...). The reason is that iOS does not yet include an implementation of Service Workers, which are needed for push notifications to work. It is currently in development, though: https://webkit.org/status/#specification-service-workers
And it is already working on Safari for OS X: https://developer.apple.com/notifications/safari-push-notifications/
Meanwhile, you will have to settle for a native implementation of your app, at least on iOS. My recommendation would be using what you have developed for web and making it a hybrid app through the use of Apache Cordova.
You can also regularly check if support is available here: