The sets $V_\lambda$ for limit ordinals $\lambda>\omega$ are precisely the models of second-order Zermelo set theory (including foundation scheme, infinity, and choice) plus the cumulative hierarchy axiom CHA, which asserts that every set $x$ is in some $V_\beta$, where this is a set for which there is well-ordered sequence $\langle V_\alpha\mid\alpha\leq\beta\rangle$ obeying the recursive definition of the cumulative hierarchy. 

It is easy to see that every $V_\lambda$ satisfies that theory, and conversely, if a model satisfies this theory, then because we have the second-order separation axiom in second-order Zermelo, it will be correct about power sets, and so the cumulative hierarchy that it builds will be correct. So the model will be (isomorphic to) $V_\lambda$ for some limit ordinal $\lambda>\omega$.

This theorem was observed by an undergraduate student of mine Donghui Jia in Oxford last term, who wrote on it emphasizing the analogy with Zermelo's quasi-categoricity theorem, in which Zermelo proved that the models of second-order ZFC are precisely $V_\kappa$ for inaccessible cardinals $\kappa$. The theorem I mentioned above is the corresponding result for the weaker Zermelo theory, when augmented with the cumulative hierarchy axiom.

Meanwhile, you had asked for consequences of replacement that hold in every $V_\lambda$, and the cumulative hierarchy axiom is one. Mathias constructed supertransitive models of Zermelo set theory in which $V_\omega$ does not exist. Modifications of the Mathias slim-model technique allow one to construct models of Zermelo set theory where the first omitted $V_\lambda$ is for any desired limit ordinal $\lambda$, even though the model has order-types much exceeding $\lambda$ and sets of rank higher than $\lambda$. I believe also that there are models of Zermelo set theory that do not satisfy transitive containment---that is, not every set has a transitive closure; this holds in every $V_\lambda$, but one uses replacement to prove it. And Mathias has other crazy models of Zermelo set theory, where forcing extensions can have new larger ordinals not in the ground model, but this never happens over any $V_\lambda$. 

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**Warning.** To get the CHA, it isn't quite enough to say that for every ordinal $\beta$ there is a set $V_\beta$ arising from a sequence $\langle V_\alpha\mid\alpha\leq\beta\rangle$ obeying the recursive definition of the cumulative hierarchy. The reason is that the ordinals might run out before you expect, and perhaps you have $V_\alpha$ for all the ordinals $\alpha$ that exist, but there are still more sets of higher rank, whose ordinals do not exist.