Elon Musk and Jeff Bezos have become the two wealthiest men in the world thanks to their breakthrough technology companies. But Mr. Musk, through SpaceX, has long held a big lead in the pursuit closest to their hearts: colonizing space.

Now, their space race is entering a new era as Mr. Bezos approaches two milestones that, if successful, could chip away at Mr. Musk’s dominance.

Mr. Bezos’ rocket company, Blue Origin, is planning to soon launch New Glenn, a massive rocket that will compete directly with SpaceX’s reusable Falcon 9 rockets. Then Amazon — founded by Mr. Bezos, who serves as its board’s executive chair — is preparing to deploy Project Kuiper, the company’s network of satellites that will challenge SpaceX’s Starlink network in providing internet access from space in low Earth orbit.

Together, New Glenn and Project Kuiper are perhaps the most ambitious attempt in years to take on SpaceX’s tight grip on the commercial space market.

The new rocket and satellites also represent a new chapter in the longtime rivalry between the billionaires behind Amazon and Tesla. Amazon has become one of the most recognizable retail brands in the world as Tesla has dominated the market for electric cars. But Mr. Musk has been far more successful in space.

SpaceX has reshaped how rockets are built and launched since Mr. Musk started it about two decades ago. Its reusable rocket boosters have reduced the cost and increased the frequency that payloads can be shot into orbit. For transporting things like satellites as well as cargo and crew headed to space stations, governments and other customers at times have few other options.