If I have a flatpak installed locally that’s no longer listed, is there a way for me to export it so I can install it on another computer?
From Flathub? It may not be visible in the website but it is still possibly in the repo. We rarely wipe apps completely.
You can back it up using create-usb USB Drives - Flatpak documentation
It was not on flathub, its an app I’d bought. With some poking I was able to get export using the USB option though so hopeful this will work. Definitely wasn’t the command I was expecting 
Then ask the vendor for support.