Microsoft has just announced that the latest version of Edge supports the Push API.
At Pushpad we have already started working to support Edge and the browser will be supported as soon as possible.
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!