A camera will allow the shutter to activate even without any lens attached at all.
Remove the lens, press the shutter button. Does this work? If so, the the problem is likely the lens. It's possible theres a communication error on the camera side, but it's FAR more likely on the lens side because that's where all the moving parts are located.
Specifically which lens(es) are you using?
Check the pins on the camera (and these AE spring-loaded.). Make sure they spring in and out easily and that none are jammed.
If the pins or contacts are obviously dirty then you can clean with a soft cloth (don't use anything abbrasive.... The gold plating is thin and easy to rub off if cleaned too aggressively). Usually it is not dirty contacts. These problems are mostly inside the lens and the lenses often need service to repair.
Tim Campbell
5D III, 5D IV, 60Da