Since we do not say Tachanun during this period – is it also not customary to fast?
In principle, we find that one should not fast on days when tachanun is not recited. See Shulchan Aruch 568:9; Eliyah Rabbah 429:4, among other sources.
However, specifically in these days, the custom of not reciting tachanun is late, and there does not appear to be a corresponding prohibition for a private individual to fast. This emerges from the discussion of poskim concerning fasting on isru chag (see Beis Yosef, Bach, Perishah, Magen Avraham, Mishnah Berurah, siman 429), where the question focuses on isru chag, but not beyond.
Moreover, the Magen Avraham writes explicitly (492:2) that our custom is to fast after isru chag in Tishrei, though a public fast is not lechatchilah arranged for these days. The Mishnah Berurah (494) also makes specific mention of isru chag (of Succos), mentioning a debate over fasting on isru chag, but does not suggests that the days following isru chag will be included.
Therefore, for a private fast, such as on Erev Rosh Chodesh (for those who wish to fast), it is permitted to fast.