If the development is based on the cost of providing water to troops then dependence on fossil fuels is one to avoid if possible because it places further demands on the supply chain.
But solar or wind powered means they just have to deliver the plant where its needed and turn it on. I suspect they do not intend that it should sign up for a green energy contract with their local electricity supplier.
The US military usually a golden triangle requirement of good, fast and cheap, and will be happy to hit two out of three given no other choices.
This Xprize thing is a humanitarian, left-liberal project. It is described in Wikipedia as "XPRIZE is a non-profit organization that designs and manages public competitions intended to encourage technological development that could benefit mankind." So there will be no direct intention in the XPrize to make it easier or cheaper to provide water for the military, in case there is any confusion on that.
One thing I note about the XPrize is that they seem to be only interested in running costs (2 cents per litre), and they don't seem to be concerned about the initial cost of the equipment. These atmospheric water generators are not cheap - I had a quick look on Amazon to see if you could buy one there, and one is on sale at $2000 that can make 2 to 5 gallons of water per day (5 US gallons is 18.9 litres). So 2000 litres per day might require a machine, or maybe a set of machines, that costs at least tens of thousands of dollars, but presumably that cost would be paid for by some aid agency.
The military would, I assume, like the Xprize people, be interested in an off-grid solar-powered atmospheric water generator, but after a quick bit of Googling, the performance of the generators that are available today does not look very impressive. This one weighs 220 lb and can only manage up to 1.5 litres per day: