How does Elizabeth Proctor seek revenge in "The Crucible?"
For most of the play, Elizabeth Proctor is the one who is the victim of revenge, not the one who seeks it. The play revolves to some extent around the way that Abigail Williams is seeking revenge on Elizabeth and on John Proctor as well.
I suppose that you could argue, however, that Elizabeth is trying to get revenge on her husband early in the play. She is angry at him because she knows he has cheated on her. This hurts her deeply and she seems to be seeking a sort of passive revenge.
You can see this in their first scene together. He is trying to be affectionate with her but she is just a bit withdrawn and cold. This really bothers John and I think you could say that is a way in which Elizabeth is getting revenge.
I did not identify any element of revenge in Elizabeth Proctor. She made a reasonable decision to send Abigail away after her husband tells her that he has feelings for her. She does not know at the time that the girl and her husband had a relationship. When she does learn the truth she does not condemn her husband or tell on him to get him into trouble, even after she is accused of being a witch.
The only type of revenge I can think of is that by not telling on her husband he suffered greatly from his own actions. He had slept with another woman, his wife was on trial for witchcraft, and he was living under a lie. The guilt eats away at him until he can not stand it any longer and he confesses his actions. Elizabeth was aware that her telling the truth would not set her husband free. It was a decision he would have to make for himself.