Let $\mathscr C \subseteq \mathscr P (\mathbb R )$ be a family of singletons, i.e.: each element of $\mathscr C$ contains exactly one real number. Let $f: \mathscr C \to \mathbb R $ be the function that for each element $C \in \mathscr C $ selects its element, i.e. $f(C) \in C$.
Is $f$ well defined? Does one need to appeal to some form or the axiom of choice to define $f$?
You do not need to appeal to the axiom of choice.
Since there is only one choice from each singleton, you can essentially write down the function explicitly:
$$\Big\{\big\langle A,x\big\rangle\mathrel{}\Bigm|\mathrel{} x\in A\in\mathscr C\Big\}.$$
Since each $A$ is a singleton, it appears in exactly one ordered pair, so it is indeed a function whose domain is $\mathscr C$, and it is easy to see that it is a choice function.
Note that it didn't even matter that the singletons were real numbers. In the case of the real numbers, or generally any linearly ordered set, you can choose from finite subsets uniformly by noting that every finite subset has a minimal (and maximal) element in the given linear order. So we can pick that.