Let's assume that ZFC is consistent.
It is easy to show that there is an illfounded model (with compactness, e.g. here). If we have the axiom of foundation at the background level we can conclude that such a model is not isomorphic to a transitive model.
Is it possible to show that there is a model of ZFC that is not isomorphic to a transitive model without assuming foundation at the background level?
Nice question! The answer is no: Foundation is indeed needed in the metatheory.
Boffa's antifoundation axiom implies that every extensional directed graph $(X; \rightarrow)$ is isomorphic to $(A;\in\upharpoonright A)$ for some transitive set $A$. In particular, any model of ZFC is isomorphic to a transitive model - at least, that's what ZFC-Foundation+Boffa thinks.
Note that the only thing being used here is that ZFC is extensional, so the above also applies to e.g. Quine's theory NF and its variants. Even for non-extensional theories, we get a version of the above result: we get that if $T$ is a consistent $\{\in\}$-theory then $T$ has a model of the form $(A;\in\upharpoonright A)$ for some set $A$.
For a discussion of Boffa's axiom, see Chapter $5$ of Aczel's book.