Category theory itself, and thus topos theory, can be formalized in set theory(*). So, in a weak sense, everything in topos theory is already "in" set theory. Of course, set theory can be formalized in topos theory using the category Set, and so set theory is also "in" topos theory. The real question that matters is which formalization is useful for a particular purpose. For some purposes, topos theory provides a useful framework to the people who use it, and they prefer this framework over the equivalent framework where everything is rephrased in terms of set theory.
The key point about formalization in set theory is that category theory, and topos theory, are formal axiomatic systems, and any axiomatic system can be studied using set theory as a metatheory.
(*): There is a minor issue that some things in topos theory may use axioms that appear to be large cardinal axioms from the point of view of set theory. But this is not an impediment to formalizing things in set theory if we simply assume the necessary large cardinal axioms.