I don’t see any plausible way for this to happen with anything like the OTL geopolitical scheme. A space race run by Western or Soviet governments is going to favor a representative of the dominant ethnicity, even if they all adopt circa-1991 attitudes on race in 1901.
Your best bet to achieve this would be to have some African country pull a China and do a shoestring ‘Dawn of the Dragon’-style moon mission in a world without the Apollo program. What African countries are the best candidates for that? Nigeria? Congo? A South Africa that becomes majority-rule early? And how to induce an early decolonization—drag WWI on much longer?
That, or have a black Elon Musk or Jeff Bezos analogue.
EDIT: Would a Brazilian of African descent satisfy the OP?