Web Push is not available as OOB feature in Adobe Campaign as of now. But there are possibilities that it may be part of future releases.
Talking about custom implementation, for Android channel adobe used open-source FCM API to trigger the Mobile Push using registration tokens. You can refer JSSP which is used for Android nms:androidPushConnectorV2.js. You need to do reverse engineering on the same to design the web push channel.
There is no specific methodology as getting a registration token and sending web push can be implemented in multiple ways.