Five headstones were pushed over at Burngrange Cemetery in West Calder between Friday 1 May and Monday 4 May.
The incident has been reported to Police Scotland and arrangements are in place for the headstones to be reinstated.
Anyone with information on those responsible can call Police Scotland on 101, Crimestoppers anonymously on 0800 555 111 or the council on 01506 280000.
Executive councillor for the environment Tom Conn said: "It's appalling that anyone would damage headstones in this way, as it causes considerable distress to the families involved.
"One of the headstones was over five feet tall and very heavy, so it would have been dangerous for someone to push over. A child in Glasgow was killed a few years ago by a headstone in a cemetery.
"We would appeal for anyone with information on those who carried out this mindless vandalism to let the council or the police know. It can also be anonymously reported to Crimestoppers."