Though not a formally declared a Dogma of the Church, the teaching of Mary as the channel through which Jesus chooses to bestow graces upon us is a teaching of the Church through Sacred Tradition. Her mediation on our behalf began with the Incarnation, and continued throughout Jesus' earthly life, from the Wedding Feast at Cana ("Do whatever He tells you") to His passion and death ("Here is your mother; there, your son"). Mary's mediation continues still, and will for all eternity, flowing out of her love for her Son, and for us, her children.
Quoting from the VII document Lumen Gentium, Pope John Paul II said in his encyclical Redemptoris Mater,
...Mary 'by her maternal charity, cares for the brethren of her Son who still journey on earth surrounded by dangers and difficulties, until they are led to their happy homeland.' In this way Mary's motherhood continues unceasingly in the Church as the mediation which intercedes, and the Church expresses her faith in this truth by invoking Mary 'under the titles of Advocate, Auxiliatrix, Adjutrix and Mediatrix.'
Pope Leo XIII says of Mary in Octobri Mense,
The Eternal Son of God, about to take upon Him our nature for the saving and ennobling of man, and about to consummate thus a mystical union between Himself and all mankind, did not accomplish His design without adding there the free consent of the elect Mother, who represented in some sort all human kind, according to the illustrious and just opinion of St. Thomas, who says that the Annunciation was effected with the consent of the Virgin standing in the place of humanity.(5) With equal truth may it be also affirmed that, by the will of God, Mary is the intermediary through whom is distributed unto us this immense treasure of mercies gathered by God, for mercy and truth were created by Jesus Christ.(6) Thus as no man goeth to the Father but by the Son, so no man goeth to Christ but by His Mother.
Pope St. Pius X, in Ad Diem Illum Laetissimum:
13. It cannot, of course, be denied that the dispensation of these treasures is the particular and peculiar right of Jesus Christ, for they are the exclusive fruit of His Death, who by His nature is the mediator between God and man. Nevertheless, by this companionship in sorrow and suffering already mentioned between the Mother and the Son, it has been allowed to the august Virgin to be the most powerful mediatrix and advocate of the whole world with her Divine Son (Pius IX. Ineffabilis). The source, then, is Jesus Christ "of whose fullness we have all received" (John i., 16), "from whom the whole body, being compacted and fitly joined together by what every joint supplieth, according to the operation in the measure of every part, maketh increase of the body unto the edifying of itself in charity" (Ephesians iv., 16). But Mary, as St. Bernard justly remarks, is the channel (Serm. de temp on the Nativ. B. V. De Aquaeductu n. 4); or, if you will, the connecting portion the function of which is to join the body to the head and to transmit to the body the influences and volitions of the head - We mean the neck. Yes, says St. Bernardine of Sienna, "she is the neck of Our Head, by which He communicates to His mystical body all spiritual gifts" (Quadrag. de Evangel. aetern. Serm. x., a. 3, c. iii.).
I have always found the works of St. Louis Marie de Montfort on the subject of Mary to be particularly moving and helpful, especially his books True Devotion to Mary and The Secret of Mary. From The Secret of Mary:
10. (4) In the new dispensation of grace to the world through Christ, God also chose Mary to be the treasurer, the administrator and the dispenser of all his graces. The sanctifying grace which brings about our spiritual adoption as children of God, passes to us through her soul as through a crystal clear channel. Thus, everyone who is a member of the mystical Body of Christ is spiritually born both of God and of Mary - just as Jesus was. Through Christ, the salvation and grace of God is available to all men. Man's personal response of faith establishes his spiritual adoption and life in God. But, the sanctifying of graces of God gained by the sacrifice of Christ, come to us through the spiritual mediation of Mary. This is why the Church calls her the 'Mediatrix of Grace.'