PITTSBURGH—A pilot program that allowed Uber Technologies Inc. to launch its first self-driving cars in this former steel town has hit a road bump.

It isn’t about regulation, which steered Uber’s driverless program out of San Francisco. Nor is it about safety, which last month led the ride-hailing firm to temporarily suspend its self-driving testing after an accident in Arizona. Here, some top officials say Uber isn’t giving back enough to the community.

“If they are going to be involved in economic disruption, they have a moral obligation to society,” said Mayor Bill Peduto. “In a partnership, it’s not just what we can do for them.”

Mr. Peduto, a Democrat, said he plans to seek Uber’s signature on a memorandum of understanding demanding better work conditions for Uber drivers, services to some elderly residents, and improvements in fuel efficiency.

An Uber spokesman said the company hadn’t seen the memorandum. “Uber is proud to have put Pittsburgh on the self-driving map, an effort that included creating hundreds of tech jobs and investing hundreds of millions of dollars,” he said. “We hope to continue to have a positive presence in Pittsburgh by supporting the local economy and community.”

The Pittsburgh mayor’s intervention highlights a broader clash as transportation companies test new technology on public roads in a race to roll out driverless cars.

It marks the latest challenge for Uber, which has been buffeted by charges of sexism and sexual harassment (the company is investigating them), an embarrassing video of Chief Executive Travis Kalanick berating a driver (Mr. Kalanick apologized), and a lawsuit from rival Alphabet Inc. alleging that Uber stole corporate secrets for its self-driving car development (Uber says it will be vindicated).

Last week, Uber published data about its workforce, saying women comprised just 15% of its technical team, but 36% of all employees. Uber released the report in the wake of criticism from civil rights groups that had called for Uber to be more transparent about its workforce.

In Pittsburgh, Uber has set up what it calls its Advanced Technologies Center. “Its mission: To make self-driving Ubers a reality,” according to a release last September, when Uber began offering customers rides in self-driving Volvos, with a human in the driver seat ready to intervene when needed.

The company has hired about 700 people, including more than 40 engineers from Carnegie Mellon University’s National Robotics Engineering Center.

“The mayor is rightly concerned about what the relationship is between companies that are coming to leverage our assets—universities, roads and so forth—and those communities they are benefiting from,” said Grant Oliphant, president of the $1.52-billion Heinz Endowments, a philanthropic arm of the ketchup conglomerate. “It is in some ways a modern expression of the old-fashioned expectation of corporate citizenship.”

Uber has been well received by many residents and business leaders. Brian Kennedy, senior vice president for the Pittsburgh Technology Council, a trade group, said Uber’s project has made it easier to draw other high-tech companies. He added that drunken-driving incidents dropped 18% in Uber’s

Mr. Peduto, an early fan of Uber, said he’s become increasingly disillusioned in the past year.

He said Uber fought to ensure Pittsburgh wouldn’t impose a tax on ride-sharing, did a U-turn on offering free service during the pilot program, and didn’t participate in some civic causes.

For instance, Pittsburgh sought to partner with Uber to win a Smart City Challenge, which grant cities federal money for developing high-tech transportation systems. It suggested that Uber sponsor a program that would allow low-income seniors to have free access to health-care providers and help build a driverless-car test track.

“They walked away from both,” Mr. Peduto said. Pittsburgh lost the competition and the $50 million prize.

Uber, a private company valued at nearly $70 billion, believes self-driving vehicles will curb costs and limit accidents caused by human error. The technology may be necessary to help Uber one day be profitable by eliminating the cost of paying drivers and seek an initial public offering of stock.

In a note to Heinz’s Mr. Oliphant and others, Mr. Peduto said, “We have each agreed that there needs to be a new social contract that benefits workers, the environment and the cities [where] they are located. We all agree Pittsburgh can play a great role in making this happen… and must.”

Part of the allure of Pittsburgh was the robotics program at Carnegie Mellon. On display in its lobby is the first autonomous vehicle, the Terregator, which more than 30 years ago rolled along, albeit at a fairly slow speed. In 2007, the school won a prestigious Defense Department competition for developing a robotic sport-utility vehicle.

Some team members became leaders in autonomous car development in Silicon Valley, Detroit and elsewhere; some have started new ventures and returned to Pittsburgh to research autonomous vehicles.