So he's technically not actually creating any new laws... He's just telling these organizations that they should not allow concealed weapons carriers on their property, a sort of defacto no-trespassing-zone for CPL holders. Expect to see those "NO GUNS" signs you see often in some other states. Also expect more pat-downs and metal detectors at public events and festivals. So while the executive order itself is issued to city employees, public citizens and CPL holders are the ones effected in turn. If you violate the no-guns signs and policies, you will be asked to leave, and if you refuse, you will be arrested for trespassing.
It's a dirty, dirty trick and I really hope we can do something to stand against it. I've already written a letter to the Mayor's office and will follow that up with e-mails to the representatives.
As Washington CPL holders we've paid the taxes and fees and gone through all the application processes so that we would be in compliance with state law for defending ourselves. Now, using dirty tricks, the Mayor had undermined state law and our ability to carry in large portions of Seattle.