https://www.spacex.com/careers, category "software development".
But I do see on the careers page there's now a (single) Palo Alto based Software Engineer position open.
Intriguing!
The second challenge is that I am in a position to be reasonably frank when the biggest threat one can hold over me is to be fired. I have found that some managers don't appreciate employees for whom their "hire/fire power" over them doesn't give them any leverage. This combined with a general perception that I am a good manager, means that I tend to develop a followship whether or not I'm in a leadership role. This can lead to "issues" no matter how hard I try to prevent them.
Or does the "eligible to obtain the required authorizations from the U.S. Department of State" apply?
If I were you, I'd read up on ITAR qualification and, if you think you would be eligible and are highly qualified, just apply.