No. Even if each LED was a good light sensor (it's not good) the control circuitry is output-only. And you need a lens, or else every pixel receives the light from all around it, and reads just the average brightness of the room.
The "D" in LED is "Diode", commonly referring to the "P-N Junction" and its capabilities.
A CMOS sensor is an array of thousands of photodiodes. So erm... diodes are _already_ used as the basis of modern digital cameras.
I guess building one out of LEDs (diodes designed to emit light, rather than receive it) is feasible but I can't imagine it'd work very well. If you build a big enough camera and calculate the right lenses, it probably would work. But it wouldn't surprise me if the proper size of such a camera was on the order of a 20-feet by 30-foot room (with a "camera sensor" the size of maybe 5-feet by 5-feet, composed of thousands of LEDs acting as terrible photodiodes).
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A proper CMOS sensor miniturizes all of that to happen on a micrometer scale, so you can shrink modern cameras down to... well... the size of our phones.
But the principles of light, sensors, and cameras in general work even with larger / inefficient objects. You just got to build a "large enough" camera to compensate for all the problems.
EDIT: And also what the other poster pointed out about the control circuitry. The circuits in a typical LED Screen are all "output only". There's no way for the information to come back / reverse the LED Screen because the circuits are already set. I'm assuming you're somehow building those circuits and have embedded them into the LEDs somehow.
