In the movie The Conjuring 2 we saw Janet playing with a Ouija Board and then leave the board under the bed. So was that the cause of the demonic possession in her house?
If the movie is based on the "real life events": yes.
Did the paranormal activity begin after they played with a Ouija board?
Yes, at least according to the real Janet Hodgson, who says that she and her sister Margaret played with a Ouija board just prior to the start of the supernatural activity. -Daily Mail Online