Journalists at a prominent legal news website voted to authorize a strike, escalating the conflict between a convulsive digital media industry and a U.S. labor movement that’s made the sector a target.

Employees of Law360, a unit of LexisNexis, voted two years ago to unionize but have been unable to secure a contract. On Oct. 23, they voted 141-11 to greenlight a work stoppage, according to the Communications Workers of America’s NewsGuild.

“It’s the most serious action that we could...