Below the surface of the Democratic campaign, Hillary Clinton and Sen. Bernie Sanders are waging a fierce battle over the party leaders who play an outsize role in picking the presidential nominee, the so-called superdelegates.

Mr. Sanders and his backers are trying to cut into Mrs. Clinton’s large lead in pledged delegates by prying away some of her superdelegates. Rank-and-file Sanders activists have taken it upon themselves to assist, using social media, emails and phone calls in ways that have created a backlash among...