No, that looks too clean. Some Streakers and Bird Engineering Bajas came with diffs stock.
The two main options you have are:
If you have money and mechanical skills, restore it!
If you don't, I'd just get it running and beat around on it.
If you can, I'd try to track down a Briggs flathead big-block. Like an 8 HP flattie(that's probably what it came with originally).
Or you could get a brand new DuroMax 440 16 HP big block. That thing would BOOGIE!

---------- Post added at 08:56 AM ---------- Previous post was at 08:54 AM ----------
Looking at it again, that kart is remarkably clean! Looks like all it really needs cosmetically is a wire wheel treatment, some paint, and a seat!