PHILADELPHIA—Special counsel Robert Mueller’s latest move briefly hijacked a closed-door meeting of state election officials and federal cybersecurity personnel here last Friday, as phones buzzed with news alerts about his indictment against Russians allegedly behind a spree of hacks before the 2016 election.

The interruption, described by several people in attendance, caught the room off guard. Some of the details in the indictment, describing the persistent efforts to compromise both Democratic Party and state election networks,...