I have a reference that says the downward Löwenheim-Skolem theorem does not occur in intuitionistic logic. In the words of the abstract "even a very powerful version of intuitionistic set theory does not yield any of the usual forms of a countable downward Löwenheim-Skolem theorem."
Charles McCarty & Neil Tennant, Skolem's paradox and constructivism. Journal of Philosophical Logic. Springer Netherlands. Issue Volume 16, Number 2 / May, 1987. https://doi.org/10.1007/BF00257838
Also page 341 of A Companion to Metaphysics By Jaegwon Kim
"...there is no intuitionistically acceptable analogue of the classical downward Löwenheim-Skolem theorem"
