From what Bugliosi wrote, Sprague insisted on using secret (illegal?) recordings of the interviews of the witnesses and voice stress analysis on the recordings. This upset the Congress persons and they decided to vote against continuing the investigation unless Sprague resigned.
A number of conspiracy advocates like to point to that so-called "first investigation" - the one led by Sprague and with people like Robert Tannenbaum and Gaeton Fonzi - as being stopped because they were uncovering CIA involvement. But if you look at what they were doing it wasn't a investigation, it was a prosecution. Fonzi himself said that he wasn't a conspiracy theorist; he admitted he was a true blue conspiracy proponent who was convinced that elements of the CIA were behind it. And Tannenbaum made all sorts of reckless claims - he once said he saw a film of Oswald with Ferrie at an anti-Castro training camp outside New Orleans; no such video has ever been produced - that turned out to be false.
I guess if you're convinced that "the CIA" was behind it, you'll pretty much accept any measures or practices, civil liberties be damned.