If you were a magical being and you were to “extract revenge” from some cauldron of calamity, maybe you’d be using your words correctly. However, for most writers plotting vengeance for their characters (or themselves?), the proper idiom is “to exact revenge.”
***insert menacing music here***
Revenge is a delicate subject. I could see how handling it properly seems like something you might do with a pipette and a beaker, but that’s just not the case.
To “exact revenge” calls back upon an old usage of the word “exact,” specifically to both demand and obtain, most commonly by force. Yikes. A bit more intimidating than pipettes, right?