The question itself suggests that a person shouldn't be allowed to tell history related to a group of which the person doesn't belong. You wouldn't question a Russian's right to talk about the history of Math, would you? You wouldn't question the right of a Chilean to talk about the moon landings and what it meant for America, would you?
It is simply ignorant, and a form of racism itself, to suggest otherwise. Just because they share a different skin color and tone does not mean a white person cannot adequately tell the history of black Americans throughout time. While the atrocities of the past were terrible, today we are moving forward and past that, and it is foolish to hold onto the past.
It is always good to hear a story from different point of views anyone should be able to tell a story. I think if the artist does his research and puts his heart and soul into the project it should not matter what his race is, but of course he has to respect the race he is telling the story about.
Yes, a person of a different race can tell the black American story if they have done enough research. There are people that paint pictures of war even though they have never experienced it. Being an artist is about expressing the emotions of the event. Because a person did not live through something does not mean that they cannot comprehend the emotion. Also, it isn't just black history. It's American history.
If they do it respectfully, I don't see a problem. I'm fine with white artists trying to capture the history of black Americans as long as they're consulting the right databases and people. If it's some kind of obvious racial spin on their part that's over the line, but art is art and if you show it for how it is there's no issue.