Here is one example where changing the logic leads to inequivalent formulations of Vopenka's principle, but it is a different kind of change in the logic than you describe.
Namely, the change has to do with how one treats classes in set theory. In Gödel-Bernays GBC set theory, it is natural to formalize it as you did, as a single assertion in GBC making a claim about every class. In ZFC, however, set theorists usually consider classes as definable classes only, and so it is natural to formalize Vopenka's principle as a scheme of assertions, one statement for each definable class (as the assertion that for any parameters to be used with that definition, if it defines an Ord-length sequence of structures, then the Vopenka statement holds for it).
Since augmenting any ZFC model with only its definable classes makes it into a GB model (one should force global choice first, if necessary, to get GBC), it might seem that the difference in these formulations wouldn't matter much. But in fact, the two formulations of VP are different, as I argued in my answer to Mike Shulman's question, Can Vopenka's principle be violated definably?. What I proved there is that there can be a model of GBC satisfying the definable version of the Vopenka principle (the scheme), but not the full version in GBC. And the same issue applies to the concept of Vopenka cardinals, giving rise to the notion of almost-Vopenka cardinals.
The end result is that the first-order formulation of VP in ZFC is strictly weaker than the second-order formulation of VP in GBC.