Laura Stewart (20) of Cumbernauld and Mhairi Convy (18) from Lennoxtown both died after being hit by William Payne’s Range Rover in Glasgow’s North Hanover Street on December 17, 2010.
The Crown Office dropped all charges against Mr Payne in 2013 but the families have now lodged a Bill of Criminal Letters with the Lord Advocate, the first step in a private prosecution.
The women, who were both studying accountancy, were struck down while going Christmas shopping during a break from Glasgow College of Commerce. Mr Payne’s Range Rover mounted the kerb and ran them over, also injuring a third pedestrian, Mark Hopwood from Cambuslang.
A fatal accident inquiry found that the tragedy was the result of Payne suffering a blackout and could have been prevented if he had disclosed his medical history.
The move comes a day after relatives of Glasgow bin lorry tragedy victims filed a similar letter in an attempt to prosecute Harry Clarke.
Lord Advocate Frank Mulholland now has one week to consider whether to approve the prosecution, and if he refuses the families can appeal to the High Court.
A spokesman for the Crown Office said: “The Lord Advocate has received a Bill of Criminal Letters and will give it due consideration.
“The Crown position on this will be made clear to the families and the court when appropriate.”
A Scottish Government spokesman added: “The decision on whether to allow a private prosecution to proceed is for the High Court however before an application can be made to the court, the lord advocate, Scotland’s senior prosecutor, must be asked whether he agrees that the prosecution should go ahead.”