The Trump administration moved Tuesday to ban rapid-fire bump stock devices like the ones used in last year’s Las Vegas massacre, submitting a new rule that makes them illegal to own or sell and will likely set the stage for long legal battles with gun rights groups and owners.

The regulation, signed and submitted by Acting Attorney General Matthew Whitaker, is expected to be formally published by week’s end and take effect in late March, senior Justice Department officials said. Owners of bump stocks will then have 90 days...