The way it is usually presented, certainly yes. As you point out it usually refers to possibly uncountable regular ordinals, which would usually mean von Neumann ordinals. Once you have the regular ordinal, say $\kappa$, then regardless of whether $\kappa$ is a von Neumann ordinal or just a well ordered set, you need to define a sequence of objects $K_\alpha$ for each $\alpha < \kappa$, and take colimits at each limit stage, and overall at the end. That kind of argument does seem to require replacement, as far as I can see. Certainly it is possible to define a sequence of sets, even of length $\omega$ whose colimit does not provably exist in the category of sets under $\mathbf{ZFC}$ minus replacement (since $V_{\omega + \omega}$ is a model of that theory, and we can define the sequence $K_n := V_{\omega + n}$).

On the other hand, there are two alternative versions of the small object argument in Swan, *W-types with reductions and the small object argument* that don't use replacement (and also don't use definitions by recursion into universes of small types, which is, roughly speaking, the type theoretic version of replacement). The first can be carried out in a topos with natural number object and satisfying the choice principle $\mathbf{WISC}$, which I believe holds for any Grothendieck topos under the assumptions of $\mathbf{ZFC}$ minus replacement. The second is specific to the case of monic generating cofibrations in presheaf toposes, but does not require $\mathbf{WISC}$, so would work in a metatheory of $\mathbf{ZF}$ minus replacement. Both arguments are for a slightly non standard definition of cofibrantly generated, and they have a different format to the usual proof using ordinals. Instead of defining a sequence of objects along an ordinal, there are just two steps, first define a $W$-type, and then either quotient it (for the first argument) or carve out a subobject (for the second).

codomainis a not a priori a set. The Hartogs theorem on wellorderings – in its original form – does not require replacement. $\endgroup$