Sabbir Hussain, 25, and Raja Hussain, 31, were found guilty of the manslaughter of Betty Laird, 88.
A third man, Shahrear Islam-Miah, 26, was cleared.
A fourth man, Muhammed Ubaidullah, 24, had already pleaded guilty to both charges and attempting to pervert the court of justice.
Mrs Laird suffered fatal spinal injuries when the Renault Kangoo in which she was a passenger was deliberately rammed in Leeds.
The gang planned to make fraudulent insurance claims after the crash, the court was told.
Mr Justice Goss said he was “shocked at their callousness”.
All four will be sentenced today.