President Donald Trump’s growing feud with Steve Bannon is threatening the former White House strategist’s leadership of the conservative Breitbart News website and upending Mr. Bannon’s plans to wage “war” on party incumbents he deemed insufficiently loyal to the White House agenda.

Mr. Bannon’s longtime benefactors, billionaires Robert Mercer and his daughter Rebekah Mercer, are actively distancing from him even before the expected release this week of a book that has roiled Messrs. Trump and Bannon’s relationship, according...