The Texas-led group of states that’s asking a federal appeals court to kill the Affordable Care Act is trying to delay oral argument in the case.

The Republican-led states told the U.S. Court of Appeals for the Fifth Circuit July 1 they need more time to prepare a supplemental brief the court requested in late June. The letter comes a week before the court is scheduled to hear arguments in the case July 9. The California-led coalition of states defending the law has opposed Texas’s request.

The court will have to act quickly. The additional briefs are due July 3....