Mrs. Putnam's character motivation comes from a variety of previously vengeful situations.
The narration at the beginning reports that she and her husband had suggested a new minister for the town, James Bayley, who happened to be Ruth's brother-in-law. The majority of the town agreed he should be hired, but a small group made sure he did not get hired. The reason for this is never cleared up in The Crucible. For Ruth's sake, this certainly gives her reason to have anger against that small faction. Many people can accept a 'no' answer when a justification is provided; but when it is unexplained, anxiety surfaces. That anxiety can quickly turn to anger or vengence.
Mrs. Putnam also seemed particularly put off by her first encounter in the play with Mrs. Nurse. Rebecca Nurse called into account her expertise as a grandmother and mother while knowing that Rebecca has lost several children. Although we find Rebecca a woman of formidable character, Ruth Putnam could have taken the comment as if Rebecca was prideful.
The Crucible never states if this group that stopped the ministry of Bayley included Rebecca Nurse, but leaving the group vague allows the mind of the reader jump to that conclusion.
Mrs. Putnam further struggles with both the Proctors and Nurses because of a land dispute her husband has with both Francis Nurse and John Proctor.
Ruth Putnam's motivation is vengeful because she feels her family has been attacked and her ability to raise children has been questioned.