Well, the simple answer is that if there is a single model, there is a proper class of models, of each cardinality possible.

But let's instead restrict to transitive models. Then the answer is considerably different. It is consistent that all the transitive models of $\sf ZFC$ have the same height, say $\alpha$. Of course, in this case $\alpha<\omega_1$ and we have that every transitive model is countable.

This means that there are at most $2^{\aleph_0}$ models, and in fact there are at least $2^{\aleph_0}$ models as well, since there is a perfect set of reals which are Cohen generic for the model.

But the work of Harvey Friedman showed that there are models of $\sf ZF$ of height $\alpha$, of cardinality $\beth_\alpha$. This means that there are plenty of intermediate models in between, and that there are many many more transitive models of $\sf ZF$ than transitive models of $\sf ZFC$.

*Harvey Friedman*, **Large models of countable height**, *Trans. Amer. Math. Soc.* **201** (1975), 227--239.