Is there a company that can handle credit card processing for my Africa-based startup?

How about looking for some online payment providers in Africa itself (it would be easier to qualify there). If you must seek a processor outside of Africa, you would most likely me looking at a 3PP (3rd Party Payment Provider) who deal in high-risk. What this entails is that your discount rate would be much higher than average and your settlement periods would be extended, with a small percentage being held in reserve for about 8-16 weeks.