First of all, I'm not well informed on this topic and the history behind it, and I apologize if anything I say is insensitive, but I'm not understanding. And I did read all the tweets in the OP.
If black people have historically been denied equal access to healthcare, why does that translate to mistrust of a universal vaccine? And why would anyone assume it's a "white" vaccine?
Personally, I feel the issue is that misinformation and lack of understanding leads to skepticism of vaccines, and misinformation and lack of understanding is more prevalent in minority groups like lower-income black communities due to unequal access to education, etc., but if the tweets are trying to say that then they aren't getting that across very well.
Edit: ok, I see a couple of posts talking about the fear of black people being used as guinea pigs essentially. I can understand that fear, definitely, and can see how it's born out of historical data, but I think lack of understanding is still the issue. And that's of course entirely on the systems built by white people denying black communities equality in education.