Wireless networks along the Texas coast suffered outages as a result of Hurricane Harvey, federal regulators said, leaving customers in some counties with limited or no cellphone service.

Rockport, Texas, near where the hurricane made landfall, was the hardest hit, according to the Federal Communications Commission. About 95% of cell sites there aren’t working, the agency said Sunday, meaning cellphone users relying on the sites can’t send or receive phone calls or data.

Of the 7,804 cell sites across the region, 320 are out of service, or about 4%. The hardest-hit areas are Aransas County, Refugio County, Calhoun County and San Patricio County.

Those counties were subject to mandatory evacuation orders, so it was unclear how many people are affected by the outages. It was also unclear which carriers stayed online and which went down, because carriers aren’t required to disclose the information publicly.

When cell towers stop working, customers can’t access the web, send calls or texts from their cellphones unless they have Wi-Fi from their cable or wireline telecom company. If only a few towers go down, service might get slower as more people connect to the remaining towers.

At least 148,565 people in the path of the hurricane were without cable or wireline service on Sunday, the FCC said.

Most cell towers have backup batteries that can last up to eight hours, and fuel generators that can take over after that. But they can still go down if they get flooded or if equipment gets blown off the tower, carriers say.

This image shows the percentage of cell sites that were out of service by county as of Sunday morning. Photo: Federal Communications Commission

Verizon Communications Inc. said it sent in extra crews to Texas and topped off fuel in each tower’s backup generators in the days leading up to the storm.

“Verizon’s network continues to perform well throughout the storm’s impacted area,” the carrier wrote in an email. “As anticipated, commercial power is out in many places throughout Texas, but backup generators are running and refueling vendors are on standby to ensure facilities continue operating.”

Verizon is offering free service to monthly subscribers in the affected area until Sept. 8.

A Sprint Corp. spokeswoman said “a relatively small number of cell sites are impacted due to commercial power outages but overall the network is holding up well.” Sprint is giving free unlimited service until Sept. 1.

A T-Mobile spokeswoman said about half of its Corpus Christi market, which includes Rockport, had some level of service degradation, a less disruptive impact than the fallout from other recent storms, including Hurricane Matthew last year.

Hurricane Harvey barreled into Texas on Friday night leaving damage in its wake. Video of the hurricane from International Space Station shows its scale. Photo: NASA

She said impassable roads were keeping crews from responding to damaged cell towers but fewer than 50 locations were affected. T-Mobile also offered free calling and texting for customers in the affected areas and those trying to reach them.

Some crews were ready to roll out new generators and equipment on Saturday but were waiting for the storm to subside before moving, she said. “Keep in mind that Harvey is still really young,” she said Saturday. “We don’t know what’s going to happen over the next week.”

An AT&T spokesman said the company’s networks in South Texas were performing well so far. “Our technicians are working to restore service to affected areas as quickly and safely as conditions allow,” he said.

Carriers recommend people keep cellphones and backup batteries charged and use text messages instead of voice calls to reduce network congestion. Landline telephones that use copper will work during power outages, but landlines that rely on new technology, known as IP, won’t.

—Drew FitzGerald contributed to this article.

Write to Ryan Knutson at ryan.knutson@wsj.com