If you parse the sentence, .....innocent people that is the stain.... makes no sense. It does not imply anyting.
I failed English three times in high school, but let me take a crack at this. I think everything is modifying the object of the sentence history.
Robert Mueller has a history
-of illicitly targeting innocent people
-that is a stain upon the legacy of American jurisprudence.
-(it is) long and sordid
I think if the stain part of the sentence was supposed to modify people and not history, then it would have to be who are instead of that is.
So the sentence structure seems correct to me. But again, I have a history of failure.
Robert Mueller has a long and sordid history of illicitly targeting innocent people, and that is a stain upon the legacy of American jurisprudence.