Wal-Mart Stores Inc. is discouraging store workers from downloading a smartphone app designed by OUR Walmart, an organization that advocates for higher pay and other benefits, as the battle between employers and labor groups increasingly shifts to social media.

The app, released on Android phones Monday, allows Wal-Mart store employees to chat among themselves and receive advice on workplace policies or legal rights, said leaders from OUR Walmart on a conference call. The group declined to say how many people had downloaded...