It’s good to clarify that Mary herself does not perform miracles; all miracles are an act of God. That said, Catholics do believe that Mary will pray for us if we ask, and that God’s miracles can be related to her intercession. This belief has its roots in the Gospel story of the Wedding at Cana, when Mary sees the needs of the young couple and brings them before Jesus. Following this, Jesus works his first miracle and turns water into wine.
There are certainly many miracles that are associated with prayers at Marian shrines or apparition sites. The Marian shrine in Lourdes, France, is one of the most famous of these. Since 1858, 68 formally recognized miracles and more than 5,000 unexplained healings have been connected to Lourdes. You can read more about these at the shrine’s official website.