Looking over the documentation for workbox.js, I am not seeing any documentation for push or push notifications.
Does workbox offer push functionality? I am speaking of the client side, not server side. Mainly subscriptions and invoking the notification api.
And if not, how would I would need to write that service worker manually. How would I integrate a manual written service worker into a existing workbox implementation?
You don't need Workbox to implement push notifications. It can be implemented directly via the ServiceWorker APIs.
You can find some good examples here:
https://serviceworke.rs/push-get-payload_service-worker_doc.html
Also note: The Push API is not currently supported on Safari : Push API
Good luck !