No, typically they can't. In platforms like Snapchat, which used to notify about screenshots in some cases, for stories it doesn't do that. Different social media platforms have different privacy policies regarding this, but in general, screenshotting a story is a private action that the story owner doesn't get notified about.
In most social media platforms, the answer is no. They cannot see if you screenshot their story. Social media companies are more concerned with protecting the privacy of the person taking the screenshot. If they were to notify every time someone took a screenshot of a story, it could lead to privacy issues for the person doing the screenshotting. Also, it could discourage users from interacting with stories if they felt their every action was being monitored.
Most of the time, no. For example, on Instagram, the platform doesn't notify the user when you take a screenshot of their story. However, there could be some third - party apps or new features in the future that might change this, but as of now, it's generally not possible for them to see.