Have been sorting out travel to/from Southern Africa (Botswana specifically) that past several days and posted what I sorted out for there, while working with a travel agent that specializes in travel to Africa from the US. Since the OP is asking about South Africa specifically:

1. RSA is currently open to US travelers. You will need a current neg PCR test for Rona to get on a flt over. Current means, results were negative within 72 hrs of arriving Jberg. In addition to the "normal" thermal image they do of your face just before you hit the immigration/passport control desks, you can also expect to possibly be given a PCR test on arrival.

2. What happens if you test positive? A LOT of the outfitters have had their facilities "certified" as Covid Quarantine facilities. Means you'd have to go directly from the airport to the outfitters place and no side trips. Once on the outfitters property, you can hunt, but only on the property contiguous to his camp. Means you can't hop from area to area and back to main camp. When you leave, you'll need to have a neg Rona test to get back on the plane to the US. PCR tests are supposedly readily available and also at the airport in Jberg. What neither I or the travel agent could sort out is IF you test positive in Jberg, and IF you have a connecting flt inside RSA if you'd be able to board the connecting flt. My money guess would be, "not" but being driven directly to camp would be allowed.

3. Current as of the last few days, is US citizens CAN return to the US from RSA if you have a neg PCR test. You MAY or MAY NOT be required to do a PCR test at the US port of entry or not and be "asked" not "required" to self quarantine at home for a week.

All this is highly flexible and can change but it's the best info I can find at this time.

There are some smoking good deals being offered in RSA now and if you're flexible you could pick up some hunts for prices that might not last too long.

There are airlines that will get you from the US to Jberg: Lufthansa, Swiss Air, Qatar Air, Turkish Air, Ethiopian Air and if Emirates is not open now, they will be very soon. Delta and Continental have CNX's all flts to Jberg and I don't remember when they will try to resume.

Again, all of this is the best I can find at this moment. Everything is constantly moving/changing.