No, they can't. Instagram doesn't provide any kind of notification or indication to the user whose IG story has been screenshotted. You can freely screenshot any public or private (if you can view it) IG story without the worry of the owner finding out through Instagram's official means. But of course, if you share the screenshot in a way that gets back to the person, they may figure it out, but that's not due to Instagram's functionality.
Instagram doesn't currently have a feature that allows someone to see if you've screenshot their IG story. So, in general, you can screenshot without the other person being notified. But keep in mind that Instagram's features can change over time, so it's always good to stay updated on their privacy policies and terms of use.
Yes, Instagram does not notify the user when you take a screenshot of a regular story. However, for some types of disappearing content like close friends stories or vanishing photos in direct messages, Instagram may have different policies in the future, but as of now, for normal IG stories, screenshotting is private.