Will ordinary bbq charcoal do a good job of removing humidity from small enclosed areas, such as under the kitchen sink?

I don’t know about charcoal, but I’ve heard that kitty litter will do a great job of de-humidifying a space. You can also buy a product (not sure what it’s called, I think it’s made of silica, I’m sure you could ask at a hardware store) that’s designed for this sort of thing. It’s the same stuff that comes sealed in those little packets in various products.

I’m looking for the cheapest solution.

Plain white rice mixed with baking soda. Can’t get cheaper than that.

A container of that crystal cat litter would make an excellent dessicant. Spread some out on a small jar lid or bowl to maximize surface area. Less likely to draw vermin than rice, too.