So we do have backups on our server, but some of the staff are not very knowledgeable with tech and sometimes make mistakes that make them lose the forms. We need to be able to get back into the screen recordings to get those forms back in case this happens.
No, I'm not saying "make backups of the forms"
The forms on the screen are digital, right?
Then what is the source of that data? How does it get displayed on the screen?
What I am suggesting is: at the very moment you get the form (=data) on the screen, you should write that data to a database. And that database should be properly backed up.
If you do that correctly (read: timestamped etc), you can always recover the data. And that is what you want at the end of the day.
If using IT means is not a feasible way, you can always go back to 1983 and use the way
@sharpuser was suggesting. To be honest I agree that is then the best way. Full time screen recording from the device itself will not help you if the device is malfunctioning. But I would really only go for this option as a last resort.