FREDERICKSBURG, Va.—The lead plaintiff in the Supreme Court battle over the health law’s insurance tax credits appears to qualify for veterans’ medical coverage, raising questions about his ability to challenge the law.

David King is one of four plaintiffs in a case before the court next month that could halt Affordable Care Act subsidies that make insurance cheaper for millions of Americans. At issue is whether the wording of the law allows the Obama administration to distribute tax credits to residents of up to 37 states...