Answer
Do PWAs need App Store approval?
No. A PWA installs directly from the browser, so it never passes through Apple App Review or Google Play review. You publish changes by deploying your website, and users get them on the next launch. Review only applies if you deliberately package the PWA into a store listing.
Why there is no review
The install comes from the browser, not a store: on iOS the user taps Share then Add to Home Screen, and on Chromium the browser offers an install prompt. Neither path involves a store submission.
Updates ship instantly
Because the app is your website, a deploy is the update. There is no binary to resubmit and no waiting for approval before users see a fix.
When review does apply
If you package the PWA with a wrapper such as PWABuilder or Trusted Web Activity to get a store listing, the normal Apple and Google review processes, developer accounts and annual fees apply again.