No need for air, since there is O2 naturally dissolved in VG and PG. Can't escape it, although there is not much. Enough to change the color, but not enough to significantly change the nic level. Freezing stops it.
It is now officially 5 years since I started storing nic in the freezer. I still have some of that vintage 2009 nic. Unchanged, as far as I can tell.
The main thing, as you noted, there is not much of dissolved oxygen to provide any troubles. And anyway, it will oxidize nicotine freezing does not stop oxidation, it slows it down. Usually temperature drop of 10 degrees Celcius reduces speed of chemical reaction by half. So, when putting nicotine in a freezer we slow down chemical processes for about 16 times compared to room temperature (increase of viscosity should not play a significant role since oxygen is already dissolved).