This is a tricky question to answer for two reasons. First of all, I'm not sure how far you are in the reading so I don't really know how much you know about these two men; second, I'm not sure I would call what Chillingworth wants "revenge." Forgive me if I reveal too much.
In chapter four of The Scarlet Letter by Nathaniel Hawthorne, We learn that Roger Chillingworth is actually Hester Prynne's long-lost husband (which means his name is actually Prynne). He has been gone for two years, and he arrives back home to discover his wife standing in a shameful public display on the scaffold--with a baby.
Certainly Chillingworth has every reason to exact revenge on both Hester and her lover, but that is not what he says when he meets with her in her prison cell. He says this to Hester:
"We have wronged each other.... Therefore, ...I seek no vengeance, plot no evil against thee. Between thee and me, the scale hangs fairly balanced. But, Hester, the man lives who has wronged us both! Who is he?"
Clearly he is willing to take at least half the blame for this shameful scenario, and he has no intention of exacting revenge on Hester. He does, however (and not surprisingly), want to know who her partner in this crime/sin is. Hester, of course, will not tell him, and Chillingworth continues.
"He bears no letter of infamy wrought into his garment, as thou dost; but I shall read it on his heart. Yet fear not for him! Think not that I shall interfere with Heaven's own method of retribution or, to my own loss, betray him to the gripe of human law. Neither do thou imagine that I shall contrive aught against his life; no, nor against his fame; if, as I judge, he be a man of fair repute. Let him live! Let him hide himself in outward honor, if he may! Not the less he shall be mine!"
Chillingworth says he does not intend to seek revenge against the man or harm him in any way, nor does he intend to get in the way of God's punishment of the man, who is undoubtedly suffering from significant guilt. Though Chillingworth wants to know who the man is, he has no explicit plans to exact revenge on him. Quite the contrary, in fact. Despite his assurances, though, we are not convinced that Chillingworth intends no harm, and neither is Hester.
The reason this information is essential to your question, of course, is that Arthur Dimmesdale is Hester's illicit lover.