Elon Musk’s SpaceX is aiming to cement its position as a leading space enterprise with a mission this week that seeks to deliver four civilians to orbit for several days and then return them to Earth.

The Inspiration4 flight, which could launch as soon as Wednesday, is a more ambitious and technically difficult one than those conducted over the summer by space companies founded by Jeff Bezos and Richard Branson, according to industry executives and consultants. It is the first taking only commercial astronauts, or those flying without government backing, to orbit, SpaceX has said.

Space Exploration Technologies Corp., the formal name for the company led by Mr. Musk and President Gwynne Shotwell, has outflanked rivals by pushing to rapidly develop its space hardware and prove that it works, people familiar with the company say.

Mr. Musk founded SpaceX in 2002, investing his own funds with a goal of eventually taking people to Mars. Along the way, the company won contracts from the National Aeronautics and Space Administration, gaining agency support during the company’s early years.

Now, nearly two decades after its founding, SpaceX has helped to reinvigorate the country’s space ambitions by orchestrating high-profile launches. Last year, the company blasted two NASA astronauts into orbit, where they docked at the International Space Station. It was the first launch with humans from the U.S. in almost a decade.

“One of the big ways SpaceX shook things up was by bringing a venture capital frame of reference to launch and innovating with a much higher risk tolerance than you would see from large, publicly traded companies,” said Carissa Christensen, chief executive of BryceTech, a data and engineering firm focused in part on space. “I think SpaceX got people’s attention by blending visionary narrative with engineering innovation.”

SpaceX has faced its share of challenges as it built its operations, including explosions, in 2015 and the following year, of Falcon 9 rockets. For a two-year stretch beginning in the middle of 2007, the company repeatedly faced cash squeezes, Mr. Musk has said.

This week, SpaceX will use its Falcon 9 and its Crew Dragon capsule—the same hardware NASA has tapped for astronaut missions—to take four people to orbit.

Jared Isaacman, the Shift4 Payments Inc. chief executive who will lead the Inspiration4 flight, said his team is confident in SpaceX’s technology and the training it has put him and fellow crew members through. That regimen has included using simulators to mimic the flight and preparing for everything from a normal mission to emergencies, he said.

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The businessman paid SpaceX an undisclosed sum for the mission, which includes a charitable component and research tasks for crew members.

Mr. Isaacman said Mr. Musk, other than for the announcement of the mission, hasn’t talked about the Inspiration4 flight during their conversations.

“This mission is a stepping stone. You have to clear this obstacle in order to do many more, you know, that are bigger and grander, and get us to the moon, Mars and beyond,” he said.

SpaceX executives weren’t available for comment.

The company, valued at more than $74 billion last April according to Pitchbook, stands out in an expanding space sector that has drawn billions of dollars in fresh investment in recent years. Based in Hawthorne, Calif., SpaceX is developing a satellite-based broadband internet service and building a moon lander for NASA, in addition to orchestrating launches.

SpaceX’s ability to develop rockets that can be used multiple times has lowered the cost of flying to space, analysts and space executives have said.

Reusable space vehicles weren’t new when SpaceX and competitors began pursuing them. NASA’s fleet of shuttles that could be flown multiple times were launched 135 times until the last flight ended in 2011, according to the agency.

“The revolutionary development from SpaceX wasn’t necessarily the technology, it was really operational. They demonstrated how you can reuse a rocket without incurring the tremendous cost of taking the whole thing apart” after a mission, said Andrew Aldrin, director of the Aldrin Space Institute at the Florida Institute of Technology.

So far this year, the company has conducted 24 of the 41 licensed launches in the U.S., according to the Federal Aviation Administration. Last year, it orchestrated 25 out of the 39 launches. More are on tap, including missions for NASA and four contracted by Axiom Space Inc., a company behind a range of commercial-space projects.

A screenshot of a NASA TV livestream shows astronauts Doug Hurley and Bob Behnken as they prepared to return to earth in a SpaceX capsule last year.

Competitors are working on their own efforts to launch humans, but SpaceX has been able to pursue more ambitious flights. For this week’s flight, Inspiration4 crew members are slated to fly almost 360 miles from Earth and travel around the planet for at least three days before splashing back down off the coast of Florida.

The mission is scheduled to launch a little more than a month after Boeing Co. , along with NASA, postponed a test flight of its own spacecraft—that is meant to ferry astronauts to the space station—due to stuck valves in the vehicle’s propulsion system. Boeing had no comment.

Virgin Galactic Holdings Inc. in July flew Mr. Branson, its billionaire founder, and five others about 54 miles up before returning to ground in a roughly one-hour trip. Blue Origin LLC, founded by Mr. Bezos, launched the former Amazon.com Inc. chief executive and three others to an altitude of 62 miles in a trip that lasted about 10 minutes.

Virgin Galactic had no comment and Blue Origin didn’t respond to a request for comment.

Blue Origin is also developing an orbital launch vehicle, but SpaceX has secured the bragging rights of taking NASA astronauts to orbit first, and possibly private ones as well should this week’s flight go off as planned.

Write to Micah Maidenberg at micah.maidenberg@wsj.com