In general, when constructing the locally ringed space $(\text{Spec}(A),\mathscr{O}_{\text{Spec}(A)})$ associated to a ring $A$ we consider the basis for the topology on $\text{Spec}(A)$ given by $\mathscr{B} = {\{D(f)\}}_{f\in A}$ define a sheaf of rings on it, and extend it in the usual way to all of $\text{Spec}(A)$. When describing this process it is often the case that one proves that for $f,g \in A$ such that $D(f) = D(g)$ one must have $A_{f} \cong A_{g}$. Hence, for each open $D(f)$ we can make a choice of $\bar{f} \in A$ such that $D(\bar{f}) = D(f)$ and set $\mathscr{O}_{Spec(A)}(D(f)) = A_{\bar{f}}$. Given a ring $A$ this process does yield a locally ringed space with the desired property.
However this construction should also yield a functor $\text{Spec}: \mathsf{Rings} \rightarrow \mathsf{L.R.S}$. What I am confused about is the following: Given that the category of rings is not small does it really make sense to "choose" for each basic open subset of each ring a localization to define as the section of our sheaf? It seems to me that this process cannot work within ZFC and would require something like the addition of global choice or a universe axiom. Thinking about this problem I believe I have found a way around this by using the saturation of manipulatively closed sets to define the structure sheaf. Indeed, since $D(f) = D(g)$ one has that $S_{f} = \{1,f,f^2,...\}$ (resp $S_g$) such that the saturations $\bar{S_f} = \bar{S}_{g}$ are equal as sets, so the localizations are literally equal and not just isomorphic. Is all of this worry on my part necessary? Apologies if I have misunderstood something obvious.
There are several equivalent ways of constructing the structure sheaf, and some of them avoid implicitly making any choices like this, which means we can safely set aside such concerns about foundations once we've proved they're equivalent.
For example, Hartshorne's Algebraic Geometry (Springer 1977, GTM vol. 52) gives the following definition in section II.2: Given a commutative ring $A$ and an open subset $U \subseteq \operatorname{Spec}(A)$, we define $\mathcal{O}_{\operatorname{Spec}(A)}(U)$ to be the set of functions $$s\colon U \to \coprod_{\mathfrak{p} \in U} A_{\mathfrak{p}}$$ such that for each $\mathfrak{p} \in U$, we have $s(\mathfrak{p}) \in A_{\mathfrak{p}}$, and there exists a neighborhood $V \subseteq U$ of $\mathfrak{p}$ and elements $a, f \in A$ such that for each $\mathfrak{q} \in V$, we have $f \notin \mathfrak{q}$ and $s(\mathfrak{q}) = a/f$ in $A_{\mathfrak{q}}$ (that is, $s$ is “locally a quotient of elements of $A$”). Hartshorne then proves in Prop. 2.2 that $\mathcal{O}_{\operatorname{Spec}(A)}(D(f)) \cong A_f$ for all $f \in A$.
Once we've given an explicit construction and proven it has the desired properties, we can essentially “forget” the details of the construction in practice, similar to how one typically works with the field of real numbers using an axiomatic characterization as the (unique-up-to-isomorphism) complete ordered field rather, rather than explicitly thinking of real numbers as (say) equivalence classes of Cauchy sequences of rational numbers.