Is there an image file format that can work as a mirror?

File format is generally irrelevant.  You would need to create an app or other program that captures an image from any available attached camera and then superimposes that captured (and likely cropped) image onto another.

So, no to what you asked.

Just use a webcam.

No. Not that kind of mirror. There can't be. That's because it isn't a matter of file format. It's a matter of monitor properties. So far, no one makes a monitor that can change the reflectivity of the screen. 

Try various software that allows you to create or manipulate images. There are over 16  million colors available. Not one of those colors acts as an actual reflector. This is true no matter what software or file format you use.

Even if you take a photograph that includes a mirror, what is shown reflected in the mirror is set at the time of exposure. It won't act as a real mirror.