Hello @user264​ , Yeap, Web-push is something that Intercom not supporting at this moment :( 

You can submit this idea in this @Product Wishlist​ group.

there still some ways to send Web-push's, You can integrate Intercom with OneSignal (Onesignal is free for 30K Subscriber).

Once you get OneSignal all you need to do is connect Intercom with OneSignal for this you can use Autopilot - https://www.autopilothq.com/integrations/intercom/onesignal/./p>

Hope this helps.