See Corollary 5.1 in The accessibility rank of weak equivalences, which shows that weak equivalences of simplicial sets are finitely accessible.

See also Theorem 4.6 in Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences, which establishes the same result in a different way.

This proves that sSet is a finitely combinatorial model category,
because it has a set of generating (acyclic) cofibrations
with compact (co)domains (compact simplicial sets are
precisely simplicial sets with finitely many nondegenerate simplices),
and its underlying category is locally finitely presentable
because it admits a set of compact generators given by all representables,
i.e., simplices.

If your goal is to prove that sSet is merely a combinatorial model category (without finiteness), then this follows much more directly and immediately from the fact that sSet is a cofibrantly generated model category (by construction), whose underlying category is a locally presentable category (just like any category of presheaves of sets on a small category). The accessibility of weak equivalences is then an automatic consequence of these facts.

$\def\Exi{{\rm Ex}^\infty}$

Finally, if one merely wants to prove the accessibility of weak equivalences directly, without referring to model structures,
this can be done as follows.
Recall the original definition of simplicial weak equivalences due to Kan:
a simplicial map $f$ is a weak equivalence if $\Exi(f)$ is a simplicial homotopy equivalence.
The functor $\Exi$ is an accessible functor between locally presentable categories because $\Exi$ is the filtered colimit of ${\rm Ex}^n$ and each of these preserves filtered colimits.
Thus, it suffices to show that simplicial homotopy equivalences
between Kan complexes form an accessible subcategory.

Indeed, by the simplicial Whitehead theorem,
simplicial homotopy equivalences between Kan complexes can be characterized by a relative-homotopy-lifting property
with respect to the set of morphisms ∂Δ^n→Δ^n.
The relative-homotopy-lifting property can be reformulated as an ordinary extension
property with respect to a certain morphism in the category of simplicial maps,
see the formula (3.3) in Dugger and Isaksen's paper Weak equivalences of simplicial presheaves.
But such a lifting property singles out an accessible subcategory of the category of simplicial maps between Kan complexes,
see the proof of Corollary A.2.6.8 in Lurie's Higher Topos Theory.