Below are the things that I discovered:
1)Vtex does provide a PWA extension that can be used .
2)The PWA app is working fine with Online connectivity and one can place orders from it .
3)Offline navigation for pages which are already loaded in past is happening
Iām not sure if you got through this but as per my findings, enable pwa link works for me to install the app and check offline storage for already visited pages as you have mentioned. There is a thread which talks about sending push notifications from admin console to app which is not resolved yet as per this github issue but talks about vendor apis.
I searched for push apis within app store and found two viz. oracle webpush and pushnews.
I was able to get push api notifications in working state on my VTEX dev workspace. I installed Onesignal app in my workspace and admin console, tried sending messages from OneSignal dashboard to my workspace and it worked by following a couple of references:
OneSignal - I used these app but you can have your own implementation by following the codebase.