Can I undertake volunteer work in South Africa on a visitor visa?

Yes. If you were going to volunteer for a lengthy period, then you'd need the volunteer visa. But for just two weeks, a tourist visa will be the easiest thing; if you're coming from the West (US/UK/Australia/Europe), you can just get your tourist visa when you get off the plane: no cost, no medical exams, and all you'll need is to show your return flight so they know you're not going to overstay your visa. Your organization is right: it's the much easier (and cheaper) way.