In another subreddit I recently saw a photo from the Memphis zoo in the late fifties, showing "negro day" - the one day of the week when only black visitors were allowed. (They were excluded every other day, of course.)
Did big businesses work the same way, with African-Americans only allowed to shop at certain times? Or were there specific "negro" businesses that served only that market?
You should check out the Coca-Cola / Pepsi divide over black consumers. Prior to Pepsi's emergence, Coke only went after white audiences, and ignored black consumers, which was pretty standard for the time.
Pepsi's main selling point was that it was a competitor to Coke that sold in bigger bottles at a cheaper price, which made it more appealing to less affluent consumers (a group that included a lot of black buyers).
Knowing that this was their big selling point, in the 1940s Pepsi created a marketing team of 12 black men for a "“negro markets” department designed to advertise and sell Pepsi to black consumers - Duke Ellington became a spokesman, and the department made a point of leaking racist statements made by Coke executives to the public.
Thanks to this effort, by the 1950s, white consumers had begun referring to Pepsi as 'nigger Coke', while Coke retained its wholesome, all-American image. Black consumers would buy Pepsi, but whites who could afford not to avoided it, and so Pepsi killed the program.
In general, big corporations avoided the black consumer market - being known as a product that was popular among black Americans was a liability that wasn't worth the dollars they had to offer.
I don't know about "big corporations," as I'm not sure how big they were as businesses, but Negro League baseball (and its constituent teams) come to mind. During the 1920s through the 1940s, the Negro Leagues were very popular as barnstorming teams, especially in communities where there was no Major League baseball, such as the rural Midwest and South.
While the audiences of Negro League baseball was not specifically blacks (and many whites did attend Negro League games), blacks did make up a large portion of the audiences and much of the marketing and bookings in the Negro Leagues was targeted toward black audiences. This became even more true after 1944, when the Major Leagues began to be integrated, and the Negro Leagues tried to focus their audiences to create a niche and remain alive.
Another example might be the Madame C.J. Walker Manufacturing Company, a company founded by the world's first black female self-made millionaire, Sarah Breedlove (aka, Madame C.J. Walker). This company, and others like it, specialized in selling hair care and beauty products to black women, and did so very successfully, for many years. Because the company didn't really cater to anyone other than black women (and eventually men, too), I don't know if it fits your definition of a "big corporation," but despite its niche, it was very successful, financially.
Sources:
Baseball's Negro Leagues, Eisenberg, Matthew
Hair Story: The Roots of Black Hair in America, Byrd and Tharpes