I do not think there are any camping grounds near Falkberget. But as you anyway need a car to go there (the buses go very seldom to/from there to the capital area, I think. Train is another possibility but that does not go very close to the crag), it should be possible to find a camp site somewhere in Kirkkonummi and drive from there to the cliff.
There are also other cliffs in the neighbourhood that are worth visiting, not least because I do not think it makes sense to spend many days in a row at Falkberget.
But as Pete said, there are some access problems right now that has to be solved in co-operation with the landowners at Falkberget. Before that is it not allowed to climb on the cliff at all. See the announcements (news) on this site for more info.