Lada cars have been a symbol of Russia’s self-reliance since they began rolling off assembly lines during the depths of the Cold War.

On Wednesday, Lada’s factory floors ground to a halt as Western sanctions deprived its parent company of the parts and supplies it needs to make cars, according to people familiar with the matter. Thousands of workers have been placed on leave.

The disruption shows how Russia’s economy is beginning to feel the bite of sanctions the West imposed on Moscow after Russian President Vladimir Putin decided to invade Ukraine. Russia’s exclusion from the Swift interbank payments system has made it hard for Russian companies to transact with suppliers. Supply routes, particularly through Ukraine, are shut and the ruble’s devaluation has made paying for parts from outside Russia much more expensive.

Such a stoppage was once unthinkable. During Soviet times, Lada’s parent company AvtoVAZ erected a giant factory on the banks of the Volga River, capable of nurturing a homegrown supply chain.

Today, however, AvtoVAZ is owned by French car maker Renault SA and the Togliatti plant relies on a Renault factory in Romania for subassembly and components. More than 20% of AvtoVAZ’s parts—from connectors to key electronics—come from outside Russia, people familiar with the matter said.

“If trade stops, AvtoVAZ stops,” said one former AvtoVAZ board member. “Putin knows that he can’t do it by himself.” It would take months or even years to get production up and running again without the support of Renault, the former board member added.

An AvtoVAZ spokesman said the company planned to idle its plants until at least March 11 due to the global chip shortage that has affected car makers world-wide. He declined to comment on the sanctions.

Workers unwilling to use their vacation days will be paid part of their monthly income while they are on leave, the spokesman said. Employees working in spare-parts production and distribution, as well as customer service, will keep a full working schedule during this period. The spokesman added the company was making every effort to resume production as soon as possible at the Togliatti plant and another factory in Izhevsk, 700 miles east of Moscow.

Russia is facing a new-vehicle shortage if car makers like AvtoVAZ remain paralyzed. Lada is Russia’s only homegrown brand with significant market share, accounting for 21% of auto sales in Russia last year, and many foreign car makers have suspended production of vehicles at their factories in Russia. The country has around 46 million passenger cars, which are an average of close to 15 years old, according to Thomas Besson, an analyst at brokerage firm Kepler Cheuvreux.

“They still need to move, whether the country is at war or not,” Mr. Besson said. “That still requires someone to make the cars. AvtoVAZ is one of the companies with the highest degree of local integration.”

Lada and AvtoVAZ are also part of Russia’s national psyche, much like General Motors Co. in the U.S. “For Russia, Lada is a symbol of the rise of the industry,” said Nikita Novikov, an editor of the automotive publication speedme.ru.

AvtoVAZ was founded in 1966 when the Soviet Union built a mammoth factory on the banks of the Volga and renamed the city that mushroomed around it after Palmiro Togliatti, the then leader of Italy’s Communist Party. The plant sprawled over 1,000 acres, more than New York’s Central Park.

In the early years, it sold all of its output domestically. Russians often had to wait years to get a car. The brand became known worldwide for its durability and enjoyed golden years with the iconic Lada Niva, a boxy vehicle that some consider a precursor to the modern SUV.

Lada cars were often unreliable but their simple designs made them easy to fix.

“I drove my own car, fixed it myself, did not take it to a garage. It was cheap in fixing, cheap in driving, simple in upkeep, and comfortable,” said Vadim Ivanov, 57, a street cleaner who lives in the village of Bolshaya Izhora outside St. Petersburg, and who has owned five older-generation Lada models.

By the mid-2000s, however, AvtoVAZ was struggling with issues familiar to many of Russia’s big firms: rampant corruption and a lack of productivity and investment. In 2007, Russia put a stake in AvtoVAZ up for sale, which was acquired by the French carmaker Renault.