According to this article (dlibra.umcs.lublin.pl/Content/21626/czas17868_30_2_2012_4.pdf), cognitive grammar is an approach to grammar which takes into accounts broad perceptual principles, including Gestalt psychology, but this approach doesn’t really reify syntax and makes it subservient to semantics. We probably could start a long conversation about whether such a statement about cognitive grammar is accurate, but I’m particularly interested in whether the phenomenon of syntax can be related at all to the principles of Gestalt psychology. I’d be interested in answers that apply specifically to the linguistic domain, or answers that take a more broad or formal approach to the definition of syntax.
Langacker uses gestalt principles to argue for many structures of conceptualization(and thereby language). I think he would argue that the mental ability(and usual tendency) of reification (conceptualizing separate entities into a whole)allows us to abstract away from specifics into higher order organizational relationships which ultimately could yield what we consider hierarchical relationships. Hierarchical relationships are obviously essential to syntax so in that way I think Langacker's work would be an example of a linguist that would possibly answer "yes" to your question.
Namely, what it tells us is that hierarchical syntax is a reflection of the more fundamental human ability to reify and process ever increasingly higher order relationships out of those simpler component parts.