The best argument that Prop 8 supporters can make is that they would be injured by the simple fact that California is not enforcing a law passed by the People. But the Supreme Court has held that "[an] asserted right to have the Government act in accordance with law is not sufficient, standing alone, to confer jurisdiction on a federal court." (Allen v. Wright, 468 U.S. 737, 754 (1984))

I think there's a good chance Judge Walker will find that the Prop 8 supporters lack standing. The only parties in the case who do have standing (the Plaintiffs and the State) have not appealed and have affirmatively said that they do not want a stay. This means that if Judge Walker finds that the Prop 8 supporters do not have standing, he also will deny the motion to stay.