The action by Jacqueline Morton's sons could pave the way for payouts to the relatives of the six people killed.
Glasgow City Council, which employed lorry driver Harry Clarke, has said it does not intend to contest the action, which will be settled by insurers.
A fatal accident inquiry found Mr Clarke lost consciousness at the wheel before the crash in December 2014.
Those who died in the city centre crash were Jacqueline Morton, 51, and Stephenie Tait, 29, both from Glasgow, Erin McQuade, 18, her grandparents Jack Sweeney, 68, and his 69-year-old wife Lorraine, from Dumbarton,  and Gillian Ewing, 52, from Edinburgh.
Their families were told in December last year that they could not launch private prosecutions against Clarke.
Ms Morton's sons and four other members of their family are named in legal proceedings at the Court of Session in Edinburgh where they are seeking compensation.
There is already an agreement between the council's insurers and the family that the action will not be defended.
Last month, Harry Clarke admitted culpably and recklessly driving a car in September 2015, despite his licence having been revoked. He is awaiting sentence at Glasgow Sheriff Court.
The family of a victim of the Glasgow bin lorry crash are to sue the council for compensation.