But the purchase was the start of a nightmare. Within months of getting the car, her son, who has autism, was stopped twice by police in their South Texas town because Carvana (ticker: CVNA), the company that sold the car to Burton, hadn't registered the vehicle in her name, she says.

It wasn't until almost six months later that Burton learned the reason for the registration delay: Carvana couldn't transfer the car's ownership to her because the company—officially, at least—hadn't owned the car.

Problems with the paperwork from Carvana's earlier purchase of the car for resale were keeping it from getting official ownership, or title, to the vehicle, a Carvana customer-service agent explained in a tense phone call late last year that Burton, of Corpus Christi, recorded.

"It was horrible," says Burton, whose son had panicked each time he was pulled over by officers suspicious of the temporary out-of-state license plates that Carvana had been issuing for the 2017 hatchback in lieu of providing permanent Texas ones. "This was my gift to my son, and they messed it up so bad."