Answering your more specific question: yes, it can be so characterized, in the following way:

 - First, start with the axioms of second-order Zermelo set theory with choice (without Replacement). Note that $V_\kappa$ is a model of this if $\kappa$ is the $\omega$th inaccessible - indeed, as long as $\lambda$ is a limit ordinal greater than $\omega$ we have that $V_\lambda$ satisfies second-order Zermelo set theory with choice.

 - Next, add "For every ordinal $\alpha$ there is an inaccessible cardinal $>\alpha$."

 - Finally, add "There is ordinal which is a limit of inaccessibles."

This characterizes $V_\kappa$ up to isomorphism.