OK, time to break out the books. ;)
Just before young Leetah heals Rayek's maimed arm, one of the villagers is encouraging her to dance the grain up out of the ground, so presumably the Sun Village has some access to grain
Savah also likened Cutter to a fighting cockrel, so they do at least know what chickens are, or maybe they gather eggs from desert-dwelling birds. I think that animal fats or oil can be used in some bread instead of eggs, but I'm not sure. Any bakers wanna field that one?