In the Carmelite tradition, Elijah and Mary are brought together most closely in the image of the cloud that forms over the sea.
1 Kings 18:42:45
 So Ahab went up to eat and to drink. And Eli'jah went up to the top of Carmel; and he bowed himself down upon the earth, and put his face between his knees.
 And he said to his servant, "Go up now, look toward the sea." And he went up and looked, and said, "There is nothing." And he said, "Go again seven times."
 And at the seventh time he said, "Behold, a little cloud like a man's hand is rising out of the sea." And he said, "Go up, say to Ahab, `Prepare your chariot and go down, lest the rain stop you.'"
 And in a little while the heavens grew black with clouds and wind, and there was a great rain. And Ahab rode and went to Jezreel.
Verse 44 is the relevant verse, and how one gets the image of Mary from that, I do not know, except that when one understands it in the way of the Medieval Carmelites, it is a most beautiful metaphor.
Mary is the cloud that rises out of the sea. The sea is saltwater, undrinkable, a vast body of water, next to which the kingdom can still thirst and die. The sea is salty, impure, an image of fallen humanity with its admixture of sin. Mary rises out of this sea, pure and perfect, laden with the water of grace that will pour out through her to all humanity--not the source of Grace herself, nevertheless the container into which all is poured until it overflows out to all people, limitless, and life-giving. Not God, but human, Mary rises from the sea, pure and Immaculate in her conception, formed as a vessel of God's grace and a place of refuge for His people.
Mary may not have made her appearance in the Old Testament, but through years of meditating and contemplating the story of Elijah, the Carmelite monks and friars came to understand this passage in a Marian sense. In so doing, they enriched the understanding of Scripture and provided another key to its depths.