It depends on what data you want to integrate.
If you want to take the screen & import it to VMix you can do so using their supplied NDI utility VMix Desktop Capture.This works over ethernet.
You can then scale the imported screen & position it in a window on the main display so that it is continuously displayed together with the main content being streamed. Alternatively, you can have it as a screen to be displayed as & when you determine
VMix has a wide variety of things & ways that they can be imported.
Although I have used VMix a lot for streaming basketball games, I haven't wanted or needed to integrate FLS yet as my scoreboard/timeclock and individual foul information are software run on separate computers & can readily be imported into VMix using VMix Desktop Capture.
I also use VMix and data i would to integrate in live stream are: complete team statistic, players stats, rebounds, points scored per player etc....... all data wich we can see on TV, :)
Mi goal is to make and to show complete game statistics during the live stream. But, I need help to do that :)
I think you'll need the paid-for version of FLS to get that data out of it.
Version 7 of FLS has a menu item for in-facility output, but it is greyed-out in the standard version.
As I don't currently intend to stream all that info', I don't have the paid-for upgrade of FLS so don't know what format the in-facility output is presented in.
I guess you'll have to get that info' from Genius Sports.
I think it is a great idea & wish you well with your efforts - I may well be looking to do the same in the future when my streaming develops further.