You'd definitely want tempered safety glass in that location. I hope you weren't hurt on the sharp shards when you fell into the door. That is why code has required safety glass for decades now. I'd contact a glazer to get a piece of safety glass installed. You're not going be able to go to the local home center to get a piece of safety glass cut. It needs to be cut and then tempered, once it is tempered it can't be cut.
It may be better to replace the door with a new steel or fiberglass door with a light (window). The window would be safety glass, and it would be thermal (2 panes separated by air or an inert gas). To DIY it, you'd probably need to go with plexiglass. I broke my garage access door a few years ago when my weed wacker kicked up a stone that hit the window. It was safety glass. Since the door was relatively new (1999 build) and a standard door, I was able to order a replacement window from Lowe's and swap it out in 20 minutes. You could possibly get a replacement like I used if the opening is standard to what is being used in doors today. I'm not sure how likely that is, since it would appear your door is from before the 1970s.