Can you photograph both sides of an ID card in a single image without using mirrors?

Yes—if the requirement is simply one final image, you can do it with multiple exposures in-camera or by compositing. One practical setup is a dark scene, a long shutter, and flash: place the card on one side of the frame, trigger the flash, turn the card over and move it to the other side, then trigger the flash again before the shutter closes. That records both sides in one photo area. A black or distant background helps keep the background from building up between flashes.

If the requirement means a single instant from one viewpoint with no mirrors or editing, then no: a flat opaque card only presents one face to the camera at a time. You’d need a trick such as mirrors, physically separating the card, or using two identical cards. So photographically, it’s possible as a single resulting image, but not as a true one-moment view of both sides of one intact card without some workaround.