Exotic Car Rental in Los Angeles claims the rapper crashed the 2016 Lamborghini Huracan Spyder on June 29 after speeding down a residential street ... then left it on the side of the road with severe damage and took off.

According to the docs, YoungBoy's agent actually rented the car on June 27 and agreed he'd be the only person to operate the Lambo and would be "personally liable for any and all damages."

Exotic Car Rental claims the car was totaled, with damages exceeding $350,000. They want YoungBoy and his agent to pay ... plus any additional costs related to the wreck-and-run.

Sources close to YoungBoy deny he had any involvement with the car accident and claim the Lambo was rented by a guy in his crew ... not his agent. We're told the reason the car was left behind is because there was an injury that required a ride to the hospital.

However, sources close to Exotic tell us the rapper was ID'd by an eyewitness on the scene.

Law enforcement is still investigating.