So basically if you can't get enough water out of the air, get it out of burning sh&t....

The problem there becomes with complete combustion your resultant gases are no longer combustable. So you must figure out a way to take the heat directly and convert it to electrical energy (to drive your dehumidifier). Meanwhile you can pass your combustion products through the chiller and pull out water at least.
If you are not burning it completely, like in this example:

... then you will gasify your wood chips by creating partial combustion products (by limiting oxygen but allowing heat to "aerosolize" the wood) which can actually feed into a compression-cycle engine that will burn the remaining partial products to completion (in an oxygen-rich environment) and power an electrical generator:

At least that is my understanding of how to use the biomass as an energy source, and potentially also gain additional water from the combustion products. Either way, if you have DRY WOOD to burn, you aren't pulling water directly out of it... you are chemically converting it to get water, and you would still need an energy source to drive your cooling system... either from the wood itself (via a biomass engine) or solar. I don't see any other way. And yes, you will get a ton of CO2 release but it is "renewable" because the trees will regrow. Heck, "coal" is also technically renewable because in a million years you will have plant matter that has died and packed into the ground form again.
Here's an old wood gasifier engine that is sort of a "lost technology" since everyone uses fossil fuels... but back in the day when gas was being rationed people were driving their cars around using this:


So if they can drive a small generator or the compressor directly with this thing, they could cycle the refrigerant this way through the system, creating the necessary cooling and also getting water both from the air, and also probably the vast majority of the water coming from the actual biomass combustion itself. So they are not "pulling water out of thin air", they are pulling water out of burning hydrocarbons and spewing out tons of CO2. I wonder how well this system would work if they could not get combustibles and had to rely only on solar and the actual humidity in the air?