"It is straightforward to check that, essentially by the same proof as for $\mathcal{L}_{\omega \omega}$, Łos’s Theorem 0.6 holds for $ \mathcal{L}_{\kappa \kappa } $ and ultraproducts by $\kappa$-complete ultrafilters."
-Kanamori, Akihiro. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Springer Monographs in Mathematics) (p.37). Springer.
The proof is of course by induction on the complexity of the formula. Now, I can see why you need $\kappa$-completeness for the case of a conjunction of less-than-$\kappa$-many formulas, but do we need it for the case of a sequence of existential quantifiers of length less than $\kappa$? It seems to me that it is not needed, but I wanted to check.