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The supply crunch in Malaysia, caused primarily by staff shortages linked to virus-control measures combined with a sharp surge in global demand, poses a new problem for the auto industry. For the first half of this year, shortages largely stemmed from companies miscalculating the pace of economic recoveries and not ordering enough parts. Now they can’t always get the parts they need because Covid-19 outbreaks are denting factory output.

“It’s a bit like a game of whack-a-mole,” said Ravi Vijayaraghavan, a Singapore-based partner at the consulting firm Bain & Co. specializing in semiconductors. “We think we have supply sorted out, and then a problem suddenly pops up somewhere else.”

Some of the world’s leading car makers including Toyota Motor Corp. , Ford Motor Co. , General Motors Co. and Nissan Motor Co. have disclosed major production cuts due largely to chip shortages from factories in Malaysia. Ford suspended work for about a week at an F-150 plant in the Kansas City, Mo., area and a Fiesta factory in Cologne, Germany because of missing parts, while Toyota said it would cut global production by around 40% in September. General Motors said it expects to make 100,000 fewer vehicles in North America in the second half of the year.

The problems in Malaysia stem from the worst Covid-19 surge the country has seen since the start of the pandemic. The nation of around 32 million people has had more than 1.6 million reported cases and 15,000 deaths to date, more than half of them this summer.

On June 1, the government imposed a nationwide lockdown to stem the spread and protect its buckling healthcare system, but it designated electronics companies as essential businesses and allowed them to operate at 60% capacity. As vaccination rates picked up, factories were eventually allowed to resume full operations, but they have been playing catch-up ever since and disruptions have persisted.

Even minor disruptions can dramatically shift output and delivery timelines. In June, the Malaysian chip maker Globetronics Technology Bhd., which assembles sensors for a U.S. smartphone maker as well as basic car components, voluntarily closed two of its factories for several days after three workers tested positive for Covid-19.

It took about four weeks to normalize deliveries, according to Chief Executive Heng Huck Lee, who had back-to-back calls with customers as he and his team tried to shuffle around orders without causing a domino effect of delays.

Mr. Heng said that employee safety was his priority and that production at the two factories was halted completely for two days to sanitize them from top to bottom and flush out the air several times over. Hundreds of workers were isolated.

“This wasn’t a disaster in terms of revenue, but it caused disruption,” Mr. Heng said. “When your workforce is lower, it kind of cascades down the line.”

Mr. Vijayaraghavan said chip manufacturing relies on a precarious model designed to keep costs low by holding minimal inventory and spreading assembly across several markets specializing in processes that are hard to relocate in a pinch. “There’s very little room for error, so whenever there’s any disruption you see it all the way through to the end product because there’s just no slack in the system,” he said.

Malaysia is a major hub for packaging, a labor-intensive process of combining basic elements into functioning components and testing them for quality before they are shipped abroad and made into end-use products. About 7% of the global supply of semiconductors goes through the country at some point, according to the U.S.-based Semiconductor Industry Association. The U.S. imports more chips directly from Malaysia than from any other country in the world, the group said.

Problems in Malaysia started last year amid factory shutdowns caused by the pandemic, though they weren’t as consequential because, at the time, global chip supply appeared more stable and demand was lower. But by early 2021, as economies started waking up, shortages were magnified by soaring demand for all manner of devices that require the chips. Then came the unexpected blows to two industry juggernauts: a fire at a major plant on the outskirts of Tokyo in March and a drought in Taiwan that slowed the chips’ water-intensive production in April. Covid-19 cases started ticking up in Malaysia around the same time, compounding the problems.

The demand has continued to rise. More chips have been needed for medical devices such as portable ultrasounds, thermometers and ventilators as healthcare systems around the world have rushed to build capacity. Prolonged lockdowns fueled demand for home appliances and personal electronics including tablets and videogame consoles. Auto makers hoped to seize on a rise in consumer spending as Western economies stepped up vaccinations and emerged from stagnation.

“Even if you’re running at 100% capacity, supply is still not enough to meet demand, and lead time will increase,” said Wong Siew Hai, president of the Malaysia Semiconductor Industry Association, a business group formed in January in response to challenges caused by the pandemic. The problem, Mr. Wong said, is that demand exceeds full capacity while most factories are still running below their potential. He estimates that pandemic-related capacity constraints translated to hundreds of millions of dollars of “missed opportunities” in the country this year.

In theory, Malaysia’s semiconductor sector should be running optimally by late August as worker vaccination rates for most companies reach the 80% threshold required to resume full operations, Mr. Wong said. In practice, factory output could be uneven for the next two or three quarters, until the entire country reaches a higher vaccination rate and transmission slows. Malaysia has fully vaccinated almost 45% of its population, according to Our World in Data.

Meanwhile, factories might be forced to shut down temporarily because of breakthrough infections among vaccinated staff, or they might have sporadic shortages related to workers being quarantined after coming into contact with an infected person, even if the workers are fully vaccinated.

Staff shortages due to quarantine requirements are a persistent problem not just on the factory floor, but for workers in related industries, such as truck drivers and cleaners. The staff shortages could cause significant disruptions in the weeks ahead, possibly longer, according to David Lacey, president of Frepenca, an industry group based in Malaysia’s industrial hub of Penang, where many semiconductor factories are located. “We’re starting to see industry capacity impacted by worker shortages, and the worker shortages are caused by quarantine rules,” Mr. Lacey said. “We need to keep the rules in sync with the situation, and the rules in place now are the same ones we had pre-vaccination.”

Chip makers said the situation remains dynamic and could be volatile into next year. A spokesperson for Infineon Technologies AG , a German semiconductor supplier that manufactures products in Malaysia, said the company expects shortages to persist into 2022, as demand remains high and significantly outstrips supply.

“Today’s semiconductor supply chain is complex to the degree that you may have to go through multiple locations in multiple countries in multiple regions just to get this one piece, this one tiny part of your final product,” said Mr. Heng, of Globetronics. “But we’ll figure out a way to overcome this. We’re all still in learning mode.”

—Chester Tay in Kuala Lumpur, Malaysia, and Jon Emont in Singapore contributed to this article.