Mark Shea in his blog, Catholic and Enjoying It, has a good Q&A:
A reader asks:
Why isn’t Mary [the Mother of God] considered the first priest since she actually brought Christ into this world?
Well, in a certain sense she is since she participates in the common priesthood of all the baptized [If she was baptized... hmmm...]. However, the short answer is: Because Christ did not ordain her to the sacerdotal priesthood.
In many respects Mary willingly participated in Christ’s perfect sacrificial self-offering in the greatest degree that any human could, and certainly greater than anyone else ever has.
But she was not chosen by Christ to be a priest in the way he chose the Apostles and their successors in the ministerial priesthood.
I do not recall any speculation in the Father about Mary having been baptized.
Her lot in the economy of salvation is privileged. I don’t think we can easily speak of her as needing precisely what all of us need, for example, the baptism which forgives the guilt of original sin, the baptism which would make the Mother of the Church a member of the Church, the baptism which would make her who gave Christ his Body a member of the Body of Christ.