I don’t know the history, and I agree it would be fascinating to hear from someone who does — but the fact that we call it a “shift register” strongly suggests it was implemented in hardware first, because “shift register” is usually a hardware term rather than a software term.

The earliest example of an LFSR I’ve seen is the Nintendo Entertainment System’s white noise generator, implemented in hardware. But many NES games used software LFSRs to generate random numbers for gameplay, so the trick was probably long-established and well-known at that time.