I think there is one main problem with what you are asking : a cheap camera probably won't come with an external triggering system.
So you could :
buy a bunch of (not so) cheap cameras (~$100 ?) with an external trigger socket, and for the trigger go the cheaper way : buy the hardware and build it yourself. You need plugs compatible with the (possibly model-specific) sockets, connect them all in parallel with some wires to a single pushbutton (you also need the schematic of the plug to know how to connect...).
buy a bunch of very cheap cameras (~$40-50) without an external trigger and "hack" them : remove the trigger button and solder/attach wires instead, and connect them to a single pushbutton like above.
Actually it may require a "two-level" pushbutton like the original one (actually a double sided temporary flip switch, or two distinct pushbuttons, or...) in order to prefocus, then take the shots.
If the camera model is the same, I don't foresee a problem with connecting all the triggers in parallel. Otherwise some electronics might become necessary (relays/optocouplers/...).
And as a final note : beware that with cheap cameras you'll get... your money's worth : fewer control options, good image quality in good lighting conditions but otherwise "maybe not", etc.