The long-secretive space ambitions of Jeff Bezos, founder and chief executive of Amazon.com Inc., suddenly are about to get a lot more public.

Blue Origin LLC, the space-exploration startup Mr. Bezos has been quietly toiling over for years, is part of a team led by Boeing Co. that is expected to soon garner a NASA contract to ferry astronauts to and from the international space station, according to people familiar with the matter.

...