You only need 4 parameters to describe the polarization at a single wavelength[1]. Naively this could be 4 parameters per color channel, so 12 channels overall. I think you could potentially need more color channels though to capture the full spectrum. But 12 channels at least looks feasible for a camera.
[1] https://en.wikipedia.org/wiki/Stokes_parameters
edit:
On second thought for dealing with reflections you might get away not capturing the "V" Stokes parameter, as you might not care about circular polarization.
edit2:
The I,Q and U parameters can be captured fully by a single polarization filter at three different rotations. This could be feasible with existing cameras with a tripod and a static subject. I wonder if this has been done before.