As far as I am aware it cannot be done. There is one courier co company in SA that is licensed to transport Ammo (Ram) , but I think you are going to find the regulations and logistics an issue.
Best would be to order ammo in advance via one of the big gun shops (e.g. Safari and Outdoor).
All of that said, I would really recommend picking a calibre that is either standard or at least fairly common here.
Ammo which is readily available in bigger cals includes .375H&H, .375Ruger, .416 Rigby, .416 Ruger, .416 Remington, .404 Jeffrey, .458WM, .458 Lott and .450 Rigby.
Personally, I don't think any client hunter needs a rifle bigger than a .416 (and most get by well with less). The only possible exception would be hunting dangerous game in very thick cover, where a .458 or .470 is a better option.