There are really two sides to this. Contracts entered into between outfitters and hunters are likely enforceable (so long as they don't deal with illegal activities - for example, a contract that requires illegal hunting likely wouldn't be enforceable anywhere), and they are likely enforceable in either the country of the hunter or that of the outfitter. That's not the same as saying which law would govern, if the contract is silent.
So if the contract doesn't say "This contract is governed by the laws of South Africa" or similar language (which usually, but not always, decides the matter of the governing law), and assuming the outfitter, and the hunt were in South Africa, and the hunter was from the US, a court would have to decide which jurisdiction had the closest connection to the subject matter of the contract. If I had to guess, I'd say South African law would likely be the governing law for enforcement of the contract, but it might be different if the contract was negotiated and entered into in the US, like at the SCI show. Then it might be US law that would govern.
Deciding which law will govern is only the beginning. You can still bring an action in South Africa to enforce a US law contract, or in the US to enforce a South African law contract, though you run the risk of the lawsuit being transferred to the other jurisdiction. Unless the contract has a clause giving exclusive jurisdiction to the courts of a particular country, and there's no public policy reason not to give effect to that. All clear so far?
So if the outfitter is mad at the US client and wants to sue him, he could launch a lawsuit in South Africa, and get a judgement, whether based on US law or South African law, and then he'd have to "register" that judgement in the US if he wanted to enforce it against the US hunter. It might have been smarter to have brought the lawsuit in the US in the first place, because that's where the other side's assets are. Equally, if the hunter is mad at the outfitter, and brings a lawsuit in the US, and wins, he'll have to register that judgement in South Africa, and try to enforce it there, because that's where the outfitter's assets likely are. Crystal clear?
The bottom line is this, in my view. A contract is a good idea because it makes both parties turn their minds to the main points of their agreement, and if they've agreed on those points, then if something goes wrong, decent people will do the right thing, without having to go to court. If one side isn't prepared to do the right thing, then you can sue, but it will almost never be worth it - the costs will outweigh the eventual recovery, unless we're talking a Lord Derby or a Mountain Nyala hunt.
Don't blame me, you asked.