Most of the roughly 21 million merchants that use PayPal Holdings Inc.’s services to sell things over the internet are inoffensive types: mom-and-pop shops on eBay Inc.’s marketplace, and big retailers like Target Corp. and Apple Inc.’s iTunes store.

Then there’s Infowars and Gab.com, websites associated with far-right-wing individuals and groups that have been accused of promoting hate speech. Each of the websites was notified this past fall that PayPal would stop processing payments for their affiliated stores or donation...