Roughly speaking, replacement is needed to justify constructions by transfinite recursion where the objects being constructed don't live in some codomain that is fixed ahead of time. In the case of the small object argument, you can write down a priori bounds on the sizes of the objects you will need to build, so replacement is not needed.

@PeterScholze I think this is related to my earlier remark, that certain maneuvers in the setting of condensed mathematics/use of the proetale topology involve strictly more set theory than is really needed in HTT (but still far less than the full strength of ZFC). Roughly speaking, for the latter one just needs to know that for any $\kappa$ one can choose an appropriately bigger $\tau$ once and for all, but the former sometimes requires iterating cardinality-raising constructions like Stone-Cech compactification infinitely many times.

@PeterScholze ... more about choosing $\tau$ appropriately; roughly speaking it needs to be "large enough compared to $\kappa$" (to have a tight dictionary between locally presentable categories and small categories that approximate them, say). If $\kappa$ is strongly inaccessible you can take $\tau=\kappa$, and then think only about one parameter. And I'd argue that this is the main feature that you want: the fact that this also guarantees that $V_{\kappa}$ models ZFC is a red herring.

@PeterScholze To phrase it differently: maybe one should think there are two cardinal parameters: $\kappa$ (bounding the size/complexity of small categories from above) and $\tau$ (bounding the size of certain big categories from below). A reflection principle guarantees you that you can choose $\kappa$ so that the small categories have expected properties (closed under operations, etcetera). But for properties you care about this is probably obvious to begin with, and don't require a careful choice of $\kappa$. The cardinality estimates appearing in the stacks project (and elsewhere) are...

@MonroeEskew: Maybe "category theory" is too broad, let's replace it by "the contents of my book". I feel 100% confident of the assertion of my first answer in the following form: every assertion in my book that concerns small $\infty$-categories can be proven in ZFC without anything stronger that $\Sigma_{15}$-replacement. Having thought about it a bit, I'm now tempted to guess that you wouldn't need any form of replacement, but I'm less sure of that.

Gabe: Yes, one does not need anything better than "$\Sigma_{15}$-universes" (and actually far far less). The whole "universe" business is just to avoid explicitly spelling out the (much weaker) closure properties that cardinals need to have to make certain arguments go through.

Peter: With a little bit less confidence, I'll say that I think the amount of "essential" set theory in my book is roughly the same as the amount needed for Grothendieck's construction of injective resolutions/Quillen's small object argument (which involves transfinite constructions but doesn't actually need replacement). I think that's actually strictly less than the amount needed for your work with Dustin (probably some free resolutions which come up there can't be formalized in ETCS?)

(Corrected version of earlier comment): ZFC allows me to construct cardinals $\kappa$ with the property that, for every theorem $\varphi$ of "ZC + $\Sigma_{15}$-replacement", ZFC will prove that $V_{\kappa}$ satisfies $\varphi$. (With the caveat that the proof is not uniform until one bounds the complexity of comprehension instances used in the proof of $\varphi$.)

Outside of set theory, I don't think you will encounter many mathematical definitions that require quantifying over the entire universe of sets. (Almost all quantifiers that appear are bounded in the sense that you are quantifying over elements of some structure which is already under discussion.)