Service worker support is currently available in Safari, but not in Apple's webview component available to developers called WKWebView.

But is kind of changing, and I believe the answer is "apps web browsers" may now enable the Service Worker entitlement, but other apps using WKWebView are still not allowed to.

This is not plainly stated anywhere, here are a bunch of tweets I am using as my basis for understanding, but I'm curious if anybody understands this better than I do.