Sending Push Notifications to iOS from PWA
You have only three main options to get push notifications working on iOS for a PWA. In both cases, you must register an App ID on Apple Developer portal, with permission to the appropriate service. For Option 1, your registered App ID must have permission to Apple Wallet. For options 2 and 3, you must have permission to Push Notifications. In both cases, you should record your Bundle ID and Team ID in case you need it later.
Option 1 (Easier): Use PassKit to set up a generic Apple Wallet pass, which can broker notifications that are very similar to native ones. Here's some documentation, and here's a working demo of how this can send push notifications to registered devices.
Option 2 (Harder): Use Firebase Cloud Messaging or a package like Node-APN to send push notifications the "proper" way, signed with a P12 or P8 key from the Apple Developer Portal. This gets tricky mainly because you need the iOS device identifier, which is only exposed to applications installed natively. I'm afraid I don't have an answer on how to get this device ID from within a PWA, and without it, this method doesn't work.
Option 3 (not a PWA): You can use an App ID with a provisioning profile and either a P12 or P8 key, similar to the previous option, but you wrap your application in Apache Cordova, and distribute it (either through the public app store, or using MDM software and via the private Apple Business Manager).
Those are your options. I have exhausted every possible avenue researching this, and I am confident that these will remain your only options through at least the next several months. It's possible we may see further support for Web Push or perhaps just a way to get the device ID from the web in the future, but until that time, this is it. There aren't any other ways to go about this presently.
Source: I architect and develop apps for major brands like Subway, Gartner, Morgan Stanley and PwC (among many others). My research is very recent, and includes direct communication with the head of WebKit at Apple, and also with one of the world's foremost PWA and iOS experts.
I just want to let you all know: Apple will support push notifications for web apps! This news was published at the WWDC2022. Apple will release Web Push with Safari 16 on macOS (Ventura) in a few months (2022) and for iOS and iPadOS in 2023.