yeah marine ply is a little overkill if it's going inside the car.
just get the lower grade stuff and coat it with something to water proof it.
another way for the draws to slide easy is rubbing wax on the surfaces that the draws will come into contact with.
in this case it shouldn't matter how rough the timber you use is it just means you'll need to put on a little more wax.

as for ilikedonuts_10:
it depends on some factors as in, how thick you want the ply to be and how heavy the fridge is, and also how far out do you intend for the fridge to be able to slide out.
i have seen actual sliders that you can buy intended for car fridges.
or you can just make your own.
what you need to worry about is just the bending moment (rotational force) that's acting on the ply at the edge of where the drawer hangs out.
reinforcements are always helpful though but you may not need it when the ply is 19mm thick and you only want it to just slide out such that you can reach it with ease.