Things You'll Need
Instructions
Fly to Veilstone City.
Walk north from the Pokemon Center to the Team Galactic building. A Galactic Grunt stands by the satellite dish outside the building.
Stand next to the grunt and press the "A" button to speak to him. He recognizes you as someone who beat him previously in battle so he panics and runs away, dropping the key in his haste.
Press the "X" button and choose the "Bag" command to view the Storage Key now in your Key Items pocket. With this key you have full access to the Galactic Headquarters building.