Yelchin, died on Sunday after his 2015 Jeep Grand Cherokee pinned him against a mailbox pillar and security fence at his home in the 3800 block of Berry Drive in Studio City, Los Angeles police said.
The 27-year-old was a rising actor best known for playing Chekov in the rebooted series.
The 2015 model-year Grand Cherokees were part of a global recall of 1.1 million vehicles announced by Fiat Chrysler in April, National Highway Traffic Safety Administration records show.
NHSA urged the recall, which involved the gear shifters. Drivers complained they had trouble telling if they had put the automatic transmissions in park. If they were not in park and a driver left the car, it could roll away.
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The agency launched a probe into this issue in August 2015 after receiving 14 such complaints.
Fiat Chrysler, which makes Jeeps, said in a statement Monday that it was investigating the crash and offered sympathies to Yelchin's friends and family.
"FCA US extends its most sincere condolences to the family and friends of Mr. Yelchin. The Company is in contact with the authorities and is conducting a thorough investigation. It is premature to speculate on the cause of this tragedy," the statement read.
As of April, Chrysler had identified approximately 700 complaints, including 212 crashes, 308 claims of property damage and 41
Investigators were looking into the position of Yelchin's gear shift at the time of the accident, authorities said.
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The actor had gotten out of the vehicle momentarily, but police didn't say why he was behind it when it started rolling.
Yelchin's friends found him after he failed to show up for an audition early Sunday.
Following an autopsy, coroner's officials ruled Yelchin's death an accident and said the cause of death was blunt traumatic asphyxia. The results of any toxicology tests would not be known for months, coroner spokesman Ed Winter said.
Yelchin's death tragically cut short the promising career of an actor whom audiences were still getting to know. "Star Trek Beyond," the third film in the series, comes out in July.
Director J.J. Abrams, who cast Yelchin in the franchise, wrote in a statement that he was "brilliant ... kind ... funny as hell, and supremely talented."
Yelchin, an only child, was born in Russia. His parents were professional figure skaters who moved the family to the United States when Yelchin was a baby. Yelchin's family requested privacy at this time.
The Associated Press contributed to this report.