A ruthless campaign of killing, extortion and kidnapping by Mexico’s powerful Zetas drug cartel has created plenty of enemies, from the Mexican government to paramilitary vigilantes to rival cartels. But now the Zetas have a new adversary: the hacker collective Anonymous.

In a video uploaded Oct. 6, an Anonymous spokesperson said that unless the Zetas release one of the group’s members, the group will reveal the photos, names and addresses of Zetas-affiliated cops and taxi drivers. (The member was allegedly kidnapped in the western coastal city of Veracruz during an “Operation Paperstorm” d...