I would just say, in as off-hand a way as possible, "Director, so-and-so asked me for a widget, and since we had 9, I sent them one."
And leave it at that. Then she knows about the widget being gone, and maybe she'll figure that's all she needs to know, and she doesn't need to inquire further. If she gets blindsided by the missing widget, she may go digging.
And--rival company? She doesn't need to hate them to be right in objecting to the sending of your organization's assets/supplies over to a rival organization!!
I might actually sic her on it, to be honest: "Director, Person Y in Dept. Y asked me for a widget, so I sent them one, since we had 9. Then, because I was on the email chain, I found out that it wasn't for Dept. Y to use, the way I thought. They sent it to Hated Rival Company. I thought you should know."
It'll keep her from being mad at you, that's for sure.