Well, lately I've taken an interest in the effects of the Void Nights upon the Khajiit people. Specifically I've been looking into what could possibly have happened to newborns during the two year period. I believe I have found an adequate answer. It is based off of the existence of fairly normal genetics, as I assumed they at least exist in a way akin to real world genetics due to the similarities between family members and such in TES. Anyways onto the theory:

Kirkbride on the Void Nights

So, according to MK, the Void nights were a eugenics experiment. Whoever performed this little test probably wanted to see how a lack of moons would effect Khajiit biologically, for whatever reason. Perhaps the perpetrators were attempting to learn something about the Lunar Lattice's link to Mundus for sinister purposes? It's too bad we'll never know. What we can try to guess is who made the moons disappear.

The Great War

The Great War states that the Thalmor claimed to have brought the moon back using "Dawn Magic". While this is simply a claim and we lack sufficient information to truly test the validity of it, it does fit in well with Kirkbride's ideas regarding the Void Nights. The Thalmor could have wanted to fidget around with the Lunar Lattice for any number of reasons and they certainly convinced the Khajiit to rethink their allies.

Words of Clan Mother Ahnissi

My final source is what actually tells us what happened to the newborn Khajiit during the Void Nights. When Azura first created the Khajiit in many different shapes. Afterwards she bound the race to the lattice. It is my belief that when the lattice is not available to shape the Khajiit as they are brought into the world then they simply developed based on the genes passed on by parents. So, during those 2 moonless years, the Khajiit simply procreated as most races do in Tamriel and their children simply resembled the parents. This seems especially likely to me as there are no stories of horrid mutants or 2 years of miscarriages, etc. If such events occurred Tamriel would certainly know more than they do. Simply giving birth to children which look like you is extremely mild in comparison.

To those of you who believe that the moons never went away at all,

If the moons were simply hidden then the Lattice would still affect the Khajiit and therefore the Khajiit would realize that the moons are present, though not visible. However, as "The Great War" illustrates, that is not the case. The Khajiit were quite distressed at the disappearance of the moons and they must have seen some difference in either their infants or their sugar, otherwise it wouldn't have mattered much at all.

So anybody want to poke some holes in my theory?

If Luna disappeared and her volume held nothing, and her gravitational effects still occurred, there would be far more trepidation and fear than if the effects went with it. It would mean that we know far, far less about things than we think we do.

"Eugenics experiment:" experiments have the null hypothesis and the alternative hypothesis, and we have no real information from which to infer either.