A few commas might make the sentence understandable:
Robert Mueller has a long and sordid history, of illicitly targeting innocent people, that is a stain upon the legacy of American jurisprudence.
“Robert Mueller has a long and sordid history of illicitly targeting innocent people that is a stain upon the legacy of American jurisprudence.”
I generally try to avoid comments on grammar, but since FReepers have made it a cause celebre herein, I opine:
The sentence is perfectly well written. As it is written, neither the addition of commas nor the word substitution (i.e., which for that) would improve it; rather, either would disimprove it.
Rewriting it with both the commas and the word substitution would make it less elegant, even if grammatically correct.
I am not a lifelong editor, but I am also old school; I scored well within the top half of the top one percentile in Standard English aptitude tests.
This sentence is written the way well-educated writers were taught to write. Grammar is no longer highly valued or well taught.