After spending the past decade eclipsed by the exploits of Elon Musk’s SpaceX, Jeff Bezos’ Blue Origin space company is over being small and done with being quiet.

Ten years ago the company consisted of the Amazon magnate himself, a few researchers and an experimental lander that fit on a flatbed truck. There was no sign outside its industrial park headquarters in a Seattle suburb, and the boss didn’t talk to the media, period.

Today...