You should be able to fire the shutter without film. A question: do you hear any noise at all when you press the shutter button? You may have encountered a somewhat common problem in MZ/ZX bodies where a plastic gear on the mirror motor cracks/breaks. The replacement part has a metal gear instead, though changing it is not super-easy. I have a ZX-5n which probably needs this surgery and it won't fire or invoke depth-of-field preview either; I hear a quiet buzzing noise when I hit the button, though I don't know if this is always the case. This camera worked intermittently when I first got it but I can no longer "make it work". Try looking at some of
these threads.