At the end I'll give you a reference to a counterexample in the paper of Meier and Ozornova. But before that, I want to talk in the other direction: besides the 3-arrow calculus that Charles Rezk mentions, there's another sense in which $W$ "almost" admits a calulus of fractions. It's a result in the same paper.
Recall the functor $\mathrm{Ex}: \mathsf{sSet} \to \mathsf{sSet}$, which is right adjoint to barycentric subdivision. There is a canonical natural transformation $1 \Rightarrow \mathrm{Ex}$, allowing to define maps $\mathrm{Ex}^n \to \mathrm{Ex}^{n+1}$ for every $n$, and most famously the colimit $\mathrm{Ex}^\infty$ was shown by Kan to be a fibrant replacement functor for the Kan-Quillen model structure on $\mathsf{sSet}$; in particular $\mathrm{Ex}^\infty X$ is a Kan complex for every simplicial set $X$ (I don't know if this extends further -- is $\mathrm{Ex}^\infty f$ a Kan fibration for every simplicial map $f$? Is $\mathrm{Ex}F$ injective-fibrant for every diagram $F$?). But moreover, a functor $F: \mathcal{C} \to \mathcal{D}$ between categories is fibrant in the Thomason model structure on $\mathsf{Cat}$ iff $\mathrm{Ex}^2 F$ is a Kan fibration (where we identify a category with its nerve).
All of this is just to motivate looking at $\mathrm{Ex}^2 \mathcal{C}$ for a category $\mathcal{C}$. Meier and Ozornova show that if $W$ is the category of weak equivalences in a model category (actually, only a substantially weaker structure called a "partial model category" is necessary), then $\mathrm{Ex}^2 W$ is a Kan complex (i.e. $W$ is Thomason-fibrant).
Why am I telling you this? Well, Meier and Ozornova also show something surprisingly (to me) clean:
Theorem: If $\mathcal{C}$ is a category, then $\mathrm{Ex}^1 \mathcal{C}$ is a Kan complex if and only if $\mathcal{C}$ admits a left calculus of fractions.
It follows that $\mathcal{C}$ admits a two-sided calculus of fractions if and only if both $\mathrm{Ex}^1 \mathcal{C}$ and $\mathrm{Ex}^1 \mathcal{C}^\mathrm{op}$ are Kan complexes. Meier and Ozornova also suggest (Question 2 at the end) that there might be notions of ``$n+1$-arrow calculus" giving clean characterizations of $\mathrm{Ex}^n \mathcal{C}$ being a Kan complex -- which in particular would recover the result that the weak equivalences in a model category admit a 3-arrow calculus, since they are Kan after applying $\mathrm{Ex}^2$.
Finally, Meier and Ozornova provide examples answering your actual question in the negative. Finite sets and injections (Example 5.1) form a partial model category whose weak equivalences do not admit a left calculus of fractions. More pertinently (Example 5.3), $\mathsf{Cat}$ with the Thomason model structure is itself a model category which does not admit a calculus of left fractions.
Note, though, that the weak equivalences in a left (resp. right) proper model category admits a left resp. right) caculus of fractions.