This is a really late answer, but the answer to your question is "No."  As a dual to the Theorem 6 that Thomas Benjamin mentions above (which I believe is a result of Stavi), Janos Makowsky [proved][1] that Vopěnka's Principle is equivalent to the statement "All logics have a compactness cardinal."




  [1]: https://www.jstor.org/stable/2273786?seq=1