Is possible that, in a choice structure $(B,C)$, $C(b) = \emptyset$, for some $b \in B$
One usually rules this out by assumption, as Mas-Colell, Whinston, and Green do in their textbook (). The goal is often to try to characterize preferences that guarantee (at least for finite choice sets) the existence of choices from a revealed preference point of view. There is also the methodological point that if you confront a decision-maker with a choice problem and chooses none of your options, then you haven't really included all options in the first place.
That being said, one can still study the mathematics of possibly empty-valued choice rules, and this has been done. One can find some material within this more general context in the book [Aleskerov, Fuad, Denis Bouyssou, and Bernard Monjardet. Utility maximization, choice and preference. Vol. 16. Springer Science &amp; Business Media, 2007.]