No. Snapchat doesn't notify the user if you screenshot their story.
Snapchat values the privacy of the viewer to an extent. When you screenshot a story on Snapchat, the person who posted the story has no way of knowing that you did so. This allows users to capture and save something they like from a story without the worry of the creator being informed.
However, it's important to note that Snapchat's policies can change over time. But currently, the answer is clear - no one can see if you screenshot their stories on Snapchat. This is different from some other features on Snapchat, like screenshotting a private snap, which may notify the sender in some cases, but for stories, it remains private.