The Nation of Islam is essentially repeating the claims made in the movie Vaxxed. This movie was made by the discredited Andrew Wakefield, author of the famous "utterly false" Lancet study on the supposed link between autism and MMR vaccines.
William Thompson himself was concerned about the presentation of data in a particular paper as discussed here. He still strong supports giving children vaccines as stated in the press release. The CDC released a statement saying that the reason the supposed vaccine/autism link was not included in the original Thompson paper was because it disappeared when the authors performed a more in depth analysis. Rather than covering this up, the CDC has made the data available, so anybody is welcome to do their own analysis if they want.
Thompson never talked to the creators of the Vaxxed movie. Rather, he had some discussions about the presentation of data with a fellow scientist - Brian Hooker. Hooker eventually published a paper pushing the autism/vaccine link in black children, but it was forcibly retracted by the journal pending an investigation due to "serious concerns about the validity of its conclusions".
Hooker recorded Thompson's conversations with him (without Thompson's knowledge). These recordings were stitched together in non-chronological ways in the movie.
The stitched together recordings created by a discredited scientist, in a movie directed by a discredited doctor, form the basis of the Nation of Islam claim. The named doctor in the claim supports vaccines in all demographics per his press release statement.