I have a system composed by amorphous A and its crystalline counterpart. Is it possible to use polarized light contrast to distinguish between the two components?
You should be able to distinguish the difference between amorphous substance and crystal using polarized light, unless your crystals have a cubic lattice. You can use a plastic plate, which will give a nice background and color your crystals. We used it to make a nice pictures of the crystals by placing them on a plastic plate.
"The simplest class of crystals are those with cubic symmetry. In a cubic crystal, all 3 crystallographic directions or axes are equivalent., and the crystal is optically isotropic. Regardless of how the light is polarized with respect to the crystal, it will experience
the same refractive index and phase delay. Therefore, any polarized light, aside from accumulating a constant phase delay, remains unchanged after traveling through a defect-free, isotropic crystal. (This is also true for amorphous substances like glass.) "
@ G. Minasov: The "crystal lattice" has nothing to do with this. You possibly mean cubic symmetry (as cited later by yourself. There is a very important difference!)? Except of hexagonal all other phases might have a "cubic" lattice so that this term you should not use.
...and your isotropy statement at the end is also sufficient for phases which only express a very small anisotropy. In other words: glass seems to be always isotropic whereas crystals have the potential of anisotropy (except for cubic symmetries). Whether you can see this under polarized light depends on the degree of anisotropy. The mysterious word is pseudosymmetry.
Also, there are some amorphous materials that exhibit stress-induced birefringence. A well known example is Prince Rupbert's Drop. I'm not sure if this is relevant to your work: https://doi.org/10.1364/AO.22.002374
For practical advice, I would consult with a geologist to learn the use of a petrological microscope (or polarizing microscope). They are often available in geology departments and used in courses and research to study crystalline thin sections. There are also books with pictures of common minerals in thin sections. EDIT: If contrast is poor, ask them whether they still have multiple oils available to use as contrast agents. Note: some of these oils are toxic.
You may check the following discussion for its possible interest concerning to your query: https://www.researchgate.net/post/What_do_the_colors_blue_yellow_etc_represent_in_birefringent_images