Is it possible to repair Face ID after original screen had been lost?

Okay you've actually got two separate issues conflated together, so let's separate them and look at them individually.

First, with regard to the reprogramming of the screen, that is strictly related to the True Tone function. When you replace a screen on later model iPhones from about the X on, the True Tone function will no longer work unless you copy a small amount of data from the original screen to the replacement with a device programmer. However, it is also possible to retrieve that data directly from the phone itself, so if the original screen is lost or non-functional, True Tone can still be restored; it's just easier to copy the data from the old screen.

Now, with regard to the other issue of your lost earpiece speaker / proximity sensor assembly; I'm afraid you're out of luck. To date there have been no successes at replacing those parts with aftermarket or even original Apple parts and having Face ID still work. The parts are paired using a proprietary encryption algorithm that only Apple has access to, so they are the only ones who can replace those parts.

@flannelist used to be an Apple technician and may know the answer to this, but it's my understanding that replacing the earpiece speaker assembly isn't a repair you can go in and ask Apple to do; they won't. However, it just might be possible to put a genuine Apple screen on your phone, break it deliberately then take it to Apple to have the screen repaired. When they do the screen replacement, they will either replace or reprogram the Face ID as part of the process and you may be able to get Face ID back that way. Alternatively, the self repair program might let you do that yourself; of course you have to buy the replacement screen from Apple so it's not like you'll be saving any money. As far as I know, that would be your only hope outside of replacing your entire Face ID system consisting of the motherboard, front camera assembly and earpiece speaker assembly with a set from a working phone.

So the TL;DR version of all this is that while you can replace your lost parts and get your True Tone function back, you can pretty much kiss your Face ID goodbye.