TRUTH OR CONSEQUENCES, N.M.— Richard Branson is scheduled to travel to the edge of space Sunday in a flight aimed at spurring a new, multibillion-dollar space-tourism industry.

The flight, originally scheduled for 9 a.m. ET, was delayed 90 minutes on Sunday because of weather overnight at the launch facility in New Mexico. The launch is now scheduled for 10:30 a.m. ET.

At that time, a highflying Virgin Galactic Holdings Inc. airplane is expected to take off from the Spaceport America facility near Truth or Consequences. The plane, called the VMS Eve, will carry the spacecraft VSS Unity, which will include Mr. Branson and five others.

The plane will take the spacecraft about 8.5 miles above Earth before releasing Unity about an hour after takeoff. The spacecraft will then rocket to an altitude of more than 50 miles.

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Virgin Galactic

The spacecraft is expected to land back in New Mexico, gliding down for a runway landing.

At the peak, Mr. Branson and other crew members will be able to experience weightlessness and peer at Earth and into space from a dozen windows in the cabin.

“After more than 16 years of research, engineering and testing, Virgin Galactic stands at the vanguard of a new commercial space industry,” Mr. Branson said earlier this month.

The flight is part of a broader push from companies and investors to develop viable businesses based on human space flight, long dominated by government space agencies with scientific and policy missions. There have been instances of private space-tourist trips in the past, such as the investor Dennis Tito’s 2001 visit to the International Space Station, but building a private industry around such travel has proven elusive for commercial enterprises so far.

A number of entrepreneurs aim to change that. Amazon.com Inc.’s founder, Jeff Bezos, has been investing in his space company, Blue Origin LLC, which plans to take him and three others to space on a company rocket later this month. Elon Musk’s Space Exploration Technologies Corp., or SpaceX, plans to fly the chief executive of a payments company to space this year.

On Friday, Blue Origin took aim at Virgin Galactic, posting a graphic on Twitter that said its New Shepard rocket has the largest windows in space, in contrast to those available on Virgin Galactic’s “high altitude airplane.”

Blue Origin said it is flying to the Karman Line, a boundary some 62 miles above Earth considered to be the start of space by many. The Federal Aviation Administration defines space as starting at 50 miles, the so-called Armstrong Line.

“New Shepard flies above both boundaries. One of the many benefits of flying with Blue Origin,” Blue Origin said in another tweet. A Virgin Galactic spokeswoman declined to comment about the tweets.

Space tourism could generate close to $4 billion in annual revenue by 2030, according to an estimate last year from UBS. Human space flight carries risks, and the vehicles developed by private space companies have been tested a fraction of the number of times compared with the planes used by airlines.

Virgin Galactic has said it plans to initiate commercial space flights that generate sales in 2022. The company founded by Mr. Branson has reported 600 reservations for future flights backed by $80 million in deposits. It reported a loss of $273 million for last year and, compared with Blue Origin and SpaceX, has a business model that is more deeply tied to the emergence of a space-tourism sector.

The company hasn’t disclosed what it will charge for tickets when it starts selling them again, but prices are likely to be out of reach for most people for some time. In the past, Virgin Galactic has sold tickets for up to $250,000 each, according to its latest annual report. Virgin Galactic interrupted marketing efforts in 2014 after a pilot died testing a company space plane.

Virgin Galactic has described Sunday’s flight—the 22nd for the VSS Unity—as part of its testing program. Now the company is focused on thinking through the experience of private astronauts it brings to space, Chief Executive Michael Colglazier said in a recent interview.

Sunday’s flight will include Mr. Branson; Beth Moses, chief astronaut instructor at Virgin Galactic; Colin Bennett, a lead operations engineer at the company; and Sirisha Bandla, a Virgin Galactic vice president focused on government affairs and research.

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“Richard’s got this long history of creating transformative consumer experiences,” Mr. Colglazier said. “He’s been dreaming about this for years, and we needed one of the members of this four-person crew in the back to really just absorb and focus on the experience side of it. He was perfectly suited to that.”

Dave Mackay and Michael Masucci will fly the VSS Unity, while CJ Sturckow and Kelly Latimer will fly the VMS Eve plane, according to the company.

During a trip in May, the spacecraft reached 55.5 miles of altitude and met the company’s technical objectives. Last month the FAA approved a full commercial space-launch license for the company.

Mr. Branson, 70 years old, is behind a business empire that includes airlines, cruise ships, hotels and gyms. Those ventures are trying to recover from the effects the Covid-19 pandemic had on travel and tourism. In recent years, Mr. Branson has made his space businesses, which include the satellite-launch company Virgin Orbit, his main focus.

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