If $\alpha$ is a limit of $2^\alpha$-supercompact cardinals,
then by the Magidor characterization of supercompactness, for each supercompact cardinal $\kappa < \alpha$, for some $\gamma < \kappa$, there is an elementary embedding $j : V_{\gamma}\to V_\alpha$ with critical point arbitrarily large below $\kappa$. Thus for each $\beta < \gamma$, there is an elementary embedding $j : V_{\gamma}\to V_\alpha$ such that $\gamma$ is between $\beta$ and the next $\alpha$-supercompact cardinal, which yields the property you asked about. 

But if $\alpha$ is the least cardinal  that is a limit of $\alpha$-supercompact cardinals, then $V_\alpha$ does not model replacement: from the perspective of $V_\alpha$, the $\omega$-many supercompact cardinals are cofinal in the ordinals. The reason is that the $2^{\alpha}$-supercompacts of $V$ are precisely the supercompacts of $V_\gamma$. Clearly the forwards implication holds, but conversely if $\delta$ is supercompact in $V_\gamma$, then $\delta$ is supercompact up to a cardinal $\kappa < \alpha$ that is $2^\alpha$-supercompact, and as a consequence, $\delta$ itself is $2^\alpha$-supercompact.