"Shortage" or "shortening"?

Shortage is a noun and shortening is a verb here.

In this case, shorten refers to the process of the days becoming gradually shorter as winter approaches. Shortage would refer to a quantity being less than a required amount. The shortening of days may very well result in a shortage of sunlight. In fact, the word "lack" can be a synonym for shortage, so they probably chose lack over shortage for style reasons.

No, You can't use it at all, but your idea, although erroneous, makes a lot of sense. Since "shortening" means making shorter then logically shortening causes a lack (of something) and vice versa if there exist a shortening of something then that is translated possibly into a deficiency, therefore a shortage. Therefore using one word or the other seems possible as the situation can be characterized by either of two concepts irremediably tied by cause and effect. This apparently correct and nice reasoning, however, does not stand the test of usage.

"Shortage" means strictly, if dictionaries do not specify it explicitly, "deficiency as it relates to a usual supply in support of a man made or animal made system". In the present case the daylight has no such recognized function of support. Of course, daylight has an action in a system at large, the system of the biosphere of which man is a part. It is not a man made system. For instance we do say that there is a shortage of ozone; ozone is being rarefied but is not an element of man made system.

As our outlook on our evironment evolves we go on to considering as man made systems new entities that apparently create contradictions, but only apparently. It wouldn't occur to anyone to speak about a shortage of whatever wild animal we can think about if not in the context of a system conceived by animals; species just become extinct; nevertheless, there is no problem talking about shortages of animals that have a function in a man made system; here is the example of a shortage of dairy cows. Beavers are wild animals and we suspect the principle to be valid for them too; however this is not quite so: beavers are an element of a portion of the ecosystem in which man has lately become an active agent of change and has therefore began to create a man made system. In this context the word "shortage" has began incidentally to be applied to beavers (ref.).

The reason for not being able to use "shortage of daylight" is usage, usage that requires the existence of a man made system (unrecorded usage should I say).

Actually, I agree with you. Shortage would make more sense, although it would imply something different from what the author meant.

Shortening refers to something getting shorter, in this case it is the day (as in "the part of the day during which we receive sunlight", opposed to night) that gets shorter. The daylight hours, however, do not get shorter. They are still 60 minutes long.

The number of hours during which we receive sunlight, however, decreases, thus causing a shortage of such hours.

Purely grammatically, within our current system of defining hours as being of a constant length, I agree that shortage makes more sense than shortening.

But...

If we read daylight hours as a descriptive form of "day" (the light part, as opposed to night), it does get shorter, and shortening of days makes grammatical sense.

Furthermore, using shortage would imply a different situation: if I experience a shortage of daylight hours, I would assume, for some reason, that I'm not getting enough daylight because, for instance, I'm staying inside too much. I would not associate that with the changing of the seasons. Lengthening and shortening of the days I do associate with the changing seasons.

Therefore, I understand the author's choice for shortening, even though the days get shorter, and not the hours.

On a side note, in ancient Rome, this sentence would have made more sense (provided it were written in Latin...). In Rome, the day was divided into twelve hours between sunrise and sunset, meaning that an hours did not always have the same length. Under that system, one could indeed observe that in fall, the hours would shorten.