Hi,
If the plastic sheets are of the same material each time, a capacitive sensor could be used.
If they are of the same color (and material) and more or less translucent to infrared, that might be a way to go.
Or you could go extremely low tech and make a mechanicel sensor (perhaps with an electronic readout).
Would it be for a adjusting the gauge of the sheets, or whar is the purpose of measuring it?