After some time working out how to do this I thought I'd put it down for posterity here - you lucky people!
I have a steam controller, so my GOG version of Fallout New Vegas needs to be launched from steam or else no controller. This is known and the net is full of how-tos. But how to get the primary launcher in Vortex to launch a non-steam game through steam?
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 gameart that comes with Vortex.
Now clcking on the little arrow in the picture in the top left (primary launcher) gives me Fallout New Vegas with my steam controller config.