Does iOS 9 mobile Safari support the Web Notifications API?
I can't find any sources pointing at new HTML5 features except for this one.
No, according to caniuse and HTML5test they don't support it.
To be honest, it isn't likely from Apples point-of-view to allow it, since it moves away from their own platform and it will have a negative impact on the battery since it has to keep the browser and its network connections alive.