I don't know if this is a WebStorm issue particularly, but its pretty strange. I have both WebStorm and IntelliJ in my taskbar. If I launch IntelliJ by clicking the tray icon, it launches fine. When I launch WebStorm, it also loads (eventually) but it takes over the IntelliJ icon in the Tray - and the WebStorm tray icon shows as unloaded.
I guess this makes some sense since IntelliJ and WebStorm share the same internals, but I have no idea how to fix it.