This same thing happened on my Ikonta.

Yes, that missing screw very well could be the reason. It holds the linkage in place that reaches from the body to the shutter. When you push the shutter button down, the linkage should rotate around that screw to trip the shutter. Without the screw, the linkage shaft won't rotate. In the upper left of the picture at the other end of the linkage, a bright metal lever sticks out of the lens/shutter body. That is your shutter button. Press that and the shutter should operate.