No. Instagram does not notify the user who posted the story if you take a screenshot of it. You can screenshot an Instagram story without the other person being aware.
As of now, on Instagram, there is no way for a person to know if you screenshot their story. Instagram is designed to allow users to view and share content freely within the app. If story creators were notified of screenshots, it could change the dynamics of how people interact with stories. For example, some users might be less likely to post certain types of stories if they knew they could be monitored in such a way. So, currently, screenshotting an Instagram story remains an anonymous action in terms of the creator being aware.
Instagram doesn't have a feature that alerts the story creator when someone screenshots their story. So, if you screenshot an Instagram story, the person who posted it won't be able to tell. This is different from some of Instagram's other features like direct messages where there are read receipts, but for stories, screenshotting is private.