Dave, would happen to know a good spot for info on these karts? I'd like to get the rear end set up on it and I'm not sure where to turn for exact specs.
As respectful as I can be, I would not spend money investing in parts with the frame you have. You can find Rupp frames and projects out there in better shape if you keep your eyes open. Your frame is far beyond "exact specs" because it needs most of it replaced.
To answer your question: You need to replace the axle, the axle bearing cassettes, bearings, sprocket, wheels, engine mount. etc. You can piece them together, or get a kit. I just did some checking around and see a few. Before you spend any money though, you need to ensure that frame is up to the task. You can find far better frames for not much money.
Go Power Sports has a live axle kit. Not sure if the link will work here, but Google it. Also, "Go Kart Alley" makes a neat kit. Here's the parts from a vendor who will sell Azusa parts as a kit. Looking at the various kits will give you a pretty good idea if what you need. No matter what, you aren't going to be able to do this without a welder and some light fabrication skills.
There are vendors around who are producing Rupp copies to exact specs. But to do an old kart to racing specs is close to $4K. I've done two of them.
If your questions and goals are more towards building a yard kart, then the above applies, but you'll need to weld on an engine mount. Otherwise, your Rupp came with a swing mount that mounted to the front on that bracket in the middle rear of the kart, on rubber grommets, and the rear of it contained bearings and it rode on the axle. You can find them too, but again, they are expensive.
The old vintage kart forum apparently crashed, and here is the new one. I'm not on Facebook, but I am sure a lot of them are there. Some of them old geezers, (grin) will gladly lend you a hand, but are probably going to recommend you do something different with that frame.
https://www.vintagekartforum.com/forumdisplay.php?fid=3 If I can be of further help, shoot me a PM, and I'll send you my email address.