One of the vehicles, carrying the civic volunteers, overturned after the driver lost control over it and hit a guard rail, police official said.
The accident took place in Char Kadirpur area in Malda police station area on Sunday night, a police officer said.
The three family members, including a couple and their daughter-in-law, were taken to a state-run hospital, where doctors declared them dead, the officer said.
PM Narendra Modi said that Rs 50,000 each would be given to those injured, according to the Prime Minister's Office.
Fourteen people died and 18 were injured after two cars were crushed by a boulder-laden lorry that fell on its side after a collision at Dhupguri.