Using Vortex to launch a GOG game with Steam controller
Create a new tool in the dashoard of Vortex, this will be the lkauncher for your game. Give it a name, the target needs to be "C:\Program Files (x86)\Steam\Steam.exe" - note, omit the quotation marks in Vortex.
The command-line options need to be of the form: "steam://rungameid/13904876691492175872" - again, omit the quotation marks.
The gameid for a non-Steam game that you have added to steam can be found by telling steam to make a shortcut on the desktop, look at its properties and you will see a complete line like the above to copy and paste.
Don't worry about an icon, because when you make the new tool primary, it still uses the large game art that comes with Vortex.
Now clicking on the little arrow in the picture in the top left (primary launcher) gives me Fallout New Vegas with my steam controller config.