First, we should look at one specific word "disciple" in the question, which creates some confusion. Jesus had scores of followers and "disciples" but there were 12 close followers of Christ, whom he specifically called, and these were the ones who often traveled with Jesus in his ministry, and the ones who were the eye-witnesses, and are portrayed in many stories in the 4 Gospels. These are the 12 apostles- Matthew, Mark, Peter, James, John, Bartholomew, etc. as distinct from the thousands of disciples, who we see also witnessed his miracles, and many of whom believed, and which became the first Christians. The word Christian means Christ follower.
We can say with certainty that Mary was one of the many disciples, but not one of the 12. She is not listed in any of the Gospels when the 12 special close disciples are listed, and she was also not listed in Acts of the Apostles with the other 12.
Now, we should look at the Biblical basis for the fact that Mary was a disciple - follower of Christ.
First and foremost was the Virgin birth. Luke chapter 2. She could easily dismiss the visit by the Angel in a dream as simply being bad pizza, but when she got morning sickness and actually had the baby, then it was clear that this was the Son of God.
Secondly, we have the account in the Gospels of the young boy Jesus, who was missing on the return journey from Jerusalem, and then they found him teaching in the Temple and the religious leaders were amazed at his wisdom. When Mary and Joseph asked him, he told them that He had to be about His Father's business.
Third, we see that Mary was present at Christ's very first miracle, and it was she who told Jesus that they were out of wine. It's clear that she didn't expect him to run to SuperTarget and get a case of wine. It's significant because she had many clues or confirmations before this, but this was the first time she saw him perform a real miracle.
Lastly, and most importantly, she was there at the foot of the cross, when he was crucified, and was listed as the only two who were the first ones to go to the tomb and who saw the risen Christ.
If there was even a tiny shred of a doubt of Christ being the Messiah before, there was certainly none when she saw him and talked to him after she witnessed death, burial and resurrection, after being in the tomb 3 days and 3 nights.