If anyone is truly serious about this, I would suggest doing the online submission from the above link. Then perhaps you might want to do a follow up to Mr. Stewart.
I was in contact with them for this position back in November. This indicates to me that they are wanting unique individuals that can stand out amongst your run of the mill candidates. The video option would definitely give you a leg up...

John Stewart <John.Stewart@spacex.com>

Me, I can't live in the hustle of Los Angeles, nor tolerate the anti gun laws of the state of Komifornia.
They do have another division outside of Waco, Texas that tests engines. They needed a CWI about 3 years ago, and you never know what can turn up if you knock on the right door.
Just thought I'd pass on the rest of the info that I have. Hope it works for one of you.