The English major in me says that there are Marian themes in anything, if you look hard enough. All joking aside, your question is a fascinating one. The first connection that came to my mind is the very name “Snow White,” with its connotations of innocence and purity. This has an obvious similarity to the Catholic beliefs that Mary was 1) conceived without sin and 2) a virgin her entire life.
I did some research and found a fascinating piece by Michael P. Duricy, called “Semiotics, Snow White, and Mary: A Mystical Rose By Any Other Name?” In it, he addresses several simliarities between Mary and the Snow White of the Disney film (keep in mind that Disney based his story on the original nineteenth-century fairytale by the Brothers Grimm, making several large revisions along the way.) Among other things, Duricy argues that the Snow White of the film represents the universal mother, as evident in her maternal care of the dwarfs and in her nurturing kindness towards the woodland animals. This resonates with the Catholic view of Mary as the mother of all. Duricy also points out some similarities between Snow White’s “sleeping death” and the terminology often used to describe Mary’s assumption into heaven at the end of her life (the passing of Mary is often called the Dormition, or “sleeping away,” by the Eastern Church).
There’s also some compelling religious imagery around the poisoned apple. In the Book of Genesis, Adam and Eve’s fall is brought about because they ate the forbidden fruit. Early Church fathers saw Mary as the inverse of Eve – while Eve disobeyed God and brought sin into the world, Mary said “yes” to God and brought forth the Savior. (This, of course, renders the Snow White/Mary link a bit problematic, given that Snow White does eat the apple – though, unlike Eve, Snow White tastes the fruit without knowing it has potentially bad consequences.)
So were these Marian themes and motifs deliberately conceived of by the Brothers Grimm and by the Disney team? It’s nearly impossible to say. Even if they were unconsciously done, though, the connections do exist for those who seek them. Duricy quotes an interview in which Disney was once asked about the deeper meaning of his work. Disney’s response? “We just try to make a good picture. And then the professors come along and tell us what we did.”