Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$.

The proof uses a lot of the power of $\sf ZF$, in particular it uses transfinite recursion which is equivalent to the axiom schema of replacement. And also the fact that we can talk about the ordinals as objects in the universe by looking at the von Neumann ordinals.

Let $\sf Z$ denote Zermelo set theory, which is obtained by removing replacement and foundation from $\sf ZF$ and adding the separation schema instead. This is a strictly weaker theory than $\sf ZF$. We know that if $\delta>\omega$ is a limit ordinal, then $V_\delta\models\sf Z$ (and in fact more, since it also satisfies foundation).

One interesting difference is that while $\sf Z+AC$ is sufficient to prove that every set can be well-ordered, it cannot prove that every set is equipotent with a von Neumann ordinal. Since, for example in $V_{\omega+\omega}$ there are only countably many von Neumann ordinals, all of which are countable, but there are uncountable sets (and one can arrange that there are uncountably many different cardinals amongst them).

Is there an inner model construction, internal to $\sf Z$, which acts like $L$ in proving the consistency of $\sf Z$ with additional axioms? If not, how about an external construction? Will the answer change if we add back the axiom of foundation?