Jacqueline Sackler was fed up. HBO’s John Oliver would soon use his TV show to pillory her family, the clan that owns Purdue Pharma, the maker of OxyContin. In a nearly 15-minute Sunday-night segment, he joined a long line of people who blamed the Sacklers in part for the nation’s opioid crisis.

Before the show aired, Ms. Sackler, who is married to a son of a company co-founder, emailed her in-laws, lawyers and advisers. “This situation is destroying our work, our friendships, our reputation and our ability to function in...