There are different kinds of dehumidifiers. The most common kind is the refrigerative dehumidifier, which is basically an air conditioner; it uses a heat pump pumps heat from its intake air into its output air, condensing water out of it in the middle, yielding liquid water.
Another kind that’s potentially simpler is the desiccant dehumidifier, which passes the air to dehumidify over a pebble bed of solid desiccant, which absorbs water from it. Eventually, though, the desiccant is full of water, and then it needs to “regenerate” the desiccant, so it connects the desiccant to a different circuit and blows hot air over it to take the water out. The hot, moist air has to go somewhere else besides into the space that it’s trying to dehumidify; typically you exhaust it outdoors.
Ideally the hot regeneration air growing more moist flows in the opposite direction from the normal cool air being dehumidified.
(Dehumidifying the air down below the point of comfort offers the opportunity to humidify it again by evaporation, which cools it evaporatively; thus, not only can you dehumidify air by cooling it (and then warming it), but you can cool air by dehumidifying it (and then humidifying it.)
One striking thing about this approach is that it can potentially be done with very primitive materials. It’s easier to do if you have plastics, microcontrollers, electric fans, silica gel, and so on, but I think you can do it entirely with stone-age materials, at least if you’re willing to accept some compromises.
Air ducting can be made from pottery, black-painted if necessary. Salt firing could make the pottery perfectly moisture-proof, and this may be desirable for some parts of the system. Pottery is fragile, and metals or plastics are a better choice where available; even lead would do.
A solar chimney is a simple way to drive airflow using solar energy. Using it to suck air through the desiccant bed implies the loss of some of the desiccated air, but I think it's possible to divert the bulk of it elsewhere.
To heat air, it can be drawn slowly through a sun-warmed duct while it’s warm.
The desiccant could be any number of common materials.
Sodium chloride salt is known since ancient times, and is probably adequately hygroscopic, but was too precious for such uses until the internal-combustion engine dropped the cost of mining by orders of magnitude. Various other salts, including sodium hydroxide, calcium chloride, sodium sulfate (mirabilite), and magnesium sulfate (Epsom salt) would also work, but none of them are very common in nature, and some of them have an inconvenient tendency to deliquesce. Calcium sulfate (gypsum, sold as the commercial desiccant Drierite) doesn't dehydrate until over 100°, and calcium hydroxide until even higher temperatures, which would require inconveniently high temperatures from the regenerating air.
Much more practical, though with a lower minimal humidity level, would be to use biomass as the desiccant. Wood chips, for example, would work, as would coffee beans, which are commonly used for this purpose in saltshakers, or cocoa beans. It’s important not to use edible materials; edible beans would attract pests unless they were poisoned. Heavily salting them would probably work without creating a hazard to humans who might consume them in desperation.
Expansive clays such as sodium bentonite would also work well, although you'd probably want to encapsulate them inside some kind of porous solid container, maybe a small unglazed prefired pottery sphere.
Control of the apparatus is achieved by opening and closing valves, which need not seal perfectly. The valves are controlled by varying temperatures, pressures, and humidities in different parts of the device. Varying humidities can be sensed with a twisted hair rope.
Nowadays, if the system has to work when there is no electrical power, we would sense temperature variation using a bimetallic strip; I don’t know what you would use in a Stone-Age situation. Perhaps you could use the lengthwise expansion of a long piece of wood referenced against ceramic; Douglas fir lengthens at 3.5 microstrains per kelvin, soda-lime glass at 8.5, and quartz at 0.33. So a 3m long piece of wood anchored to a 3m ceramic pipe would show a difference in length of 64 microns over 20°, which is probably too small to reliably activate a valve.
Tangential expansion is more like 45 microstrains per kelvin, which would bring the difference up to 0.9 mm, but that's also strongly influenced by humidity.
As an alternative to a packed column intermittently regenerated with hot air, you could use an enthalpy wheel, which continuously rotates at about 3 mHz while one side is being regenerated and the other side is being used for desiccation. It’s presumably possible to drive its rotation with a light windmill in a sufficiently tall solar chimney, or intermittently using a regular windmill.