The sets $V_\lambda$ for limit ordinals $\lambda>\omega$ are precisely the models of second-order Zermelo set theory (including foundation scheme plus infinity) plus the cumulative hierarchy axiom CHA, which asserts that for every ordinal $\beta$ there is a sequence $\langle V_\alpha\mid\alpha<\beta\rangle$ 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 should be seen as a version of Zermelo's quasi-categoricity theorem, in which he proved that the models of second-order ZFC are precisely $V_\kappa$ for inaccessible cardinals $\kappa$.
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 $\lambda$.