So I'm having some trouble understanding the existence of countable elementary submodels.
I have read and understand the Löwenheim–Skolem theorem, so given a model I understand how to build a countable submodel.
This is the point of my inquiry - does ZFC have a model that the countable elementary submodels are pulled from?
This is the unanswered detail that has been causing me to lose faith in what we're doing in my set theory class. Thanks!
Well. Maybe. But even then, it's not really enough. But don't worry, there's a workaround!
What do I mean by that? First of all $\sf ZFC$ cannot prove its own consistency. So you cannot prove in $\sf ZFC$ that $\sf ZFC$ is consistent, and so you cannot prove that it has a model to begin with. But of course if it does have a model, then it has a countable elementary submodel and it's fine.
But in most cases, e.g. in forcing, we are interested in countable transitive models. The existence of those requires an even stronger assumption than just that of countable models. The reason is that a transitive model agrees with the meta-theory about the integers and about basic number theoretic statements, like "$\sf ZFC$ is consistent". Therefore if there is a transitive model, inside lies a smaller model of $\sf ZFC$ (which may or may not be transitive).
However we can beat this problem! The reflection theorem tells us that given any finite fragment of $\sf ZFC$, there is some $V_\theta$ and some $H(\lambda)$ which satisfy this finite fragment. And since we know that $V_\theta$ satisfies all the subset schema for a limit ordinal $\theta$ (but generally not replacement), and we know that $H(\lambda)$ satisfies all the replacement schema for a regular $\lambda$ (but generally not power set), we can find transitive models of any sufficiently large fragment of $\sf ZFC$ with either one of these schemas as our needs require.
Why is this helpful? If we have a transitive model of "enough axioms of $\sf ZFC$", taking an elementary submodel we can apply the Mostowski collapse lemma to obtain a countable transitive model of the same theory. Now if we want to prove that something is consistent with $\sf ZFC$, then we can show that for every finite fragment of $\sf ZFC$ there is a model of that fragment as well the statement under consideration.
Do note, however, that in the above paragraph "every finite fragment" is taken in the meta-theory. Not internally to the universe.
So this is all very confusing at start. Which is why I'm a big fan of "for pedagogical reasons, let's just assume we have a countable transitive model of $\sf ZFC$ to begin with", and later reveal the difficulties and how to overcome them.