Microsoft has just announced that the latest version of Edge supports the Push API.
You can find the announcement on UserVoice. The new feature is also described by Microsoft on their blog and in particular in this post about service worker support on Edge.
At Pushpad we have already started working to support Edge and the browser
will be supported as soon as possible is now supported (see the updates below).
20/01/2018 Update: After some discussion, with the help of Ali A. (thanks!) from the Edge team, we have found and fixed 2 issues (1 Pushpad issue and 1 Edge issue). This means that when the next Windows Insider Preview will become available, Edge notifications should work as expected. In any case we will provide updates on this in the next weeks.
15/02/2018 Update: We confirm that Pushpad works with the new Windows build 17093. Now Edge is supported!
12/04/2018 Update: Web push notifications will become available to all Windows users – and not only to Windows Insiders – starting with the Spring Creators Update. The new version (build 17133) was expected to be released to the public on the 10th of April: however it has been postponed by two weeks, probably due to a bug.
01/05/2018 Update: Microsoft has just announced the public release of the Windows update (now called April 2018 Update). That means that all Windows users will be able to use web push notifications on Edge as soon as they get the update. The update will be rolled out to all users in the next days (probably starting on the 8th of May). However you can download the update now manually if you don’t want to wait.