While "take offence at" is used more frequently, "take offense with" is also correct and sounds more "natural" to me in some cases. Consider

"I took offence at the way that she spoke to me" versus

"I took offence with her for the way that she spoke to me."

"Take offense" usage
In Back to the Future, Marty goes back to 1955 and devises a plan for his father George McFly to win over Lorraine in order to ensure Marty's survival.

The plan involved Marty pretending to sexually assault Lorraine then George would step in and stop him. Biff meddles with Marty's plan, by kicking Marty out of the car and then he tries to rape Lorraine. George expecting to find Marty in the car, instead finds Biff and musters the courage to stop Biff and protect Lorraine.

Near the end of the movie when Marty travels back to 1985 we see that Biff is outside of Marty's house waxing George's car. Why would George employ Biff to do this job? I can't imagine Lorraine would be too happy with the idea.
