What is the role of Apostolic Succession in lifting men and women to the ranks of saints?
The short answer is that it has everything to do with bishops raising people to the rank of saints. Without it, churches can not ordain priests, consecrate bishops, celebrate the Eucharist validly or even canonize saints.
The transmission of authority, ecclesiastical powers and privileges begins with the Apostles themselves and is transmitted to their successors!
If the authority of apostolic churches comes from a direct succession of the apostles, where does the authority of non- apostolic succession church come from? They will no doubt ultimately say Jesus Christ; but then so will the others in union with their apostolic succession.
Apostolic succession is the method whereby the ministry of the Christian Church is held to be derived from the apostles by a continuous succession, which has usually been associated with a claim that the succession is through a series of bishops. Christians of the Roman Catholic, Eastern Orthodox, Oriental Orthodox, Old Catholic, Anglican, Church of the East, Moravian, and Scandinavian Lutheran traditions maintain that "a bishop cannot have regular or valid orders unless he has been consecrated in this apostolic succession." Each of these groups does not necessarily consider consecration of the other groups as valid.
Without valid apostolic succession the declaration of sainthood would be difficult in being recognized by many churches that hold on to apostolic succession.
Is there really a possibility of canonization within churches that can not prove apostolic succession?
We know from the history of the Church that in the Early Church saints were declared saints by the people as the Latin phrase suggests: Vox Populi, Vox Dei. But as the various churches became more and more religious institution, the rules governing the churches were eventually codified by church laws.
The service was held at The Christian Cathedral in Springfield, Massachusetts, on Friday by Archbishop Timothy Paul of the Holy Christian Orthodox Church. The Church is relatively new and despite the name is not part of the Orthodox family of churches.
Speaking in July 2014 after it was announced the American Baptist minister and civil rights campaigner had been nominated for the honour, Archbishop Paul said: "Dr. King was Catholic because he inspired the universal church.
"He was evangelical because of his Baptist roots, and he was charismatic. The Roman Catholic Church cannot make him a saint because he was not a Roman Catholic.
"But our church body, which has full apostolic succession, can present him to the entire Christian faith to be venerated on April 4, the date of his assassination." - Martin Luther King Jr. made a saint by American church
Not too sure that the author (Timothy Paul Baymon) of the above article could support his statement that his Church has full apostolic succession.
As for the possibility of the Roman Catholic Church not being able to canonize Martin Luther King Jr. is not an absolute impossibility, just because he was not a Roman Catholic!
In Pope John Paul II's apostolic letter Tertio Millennio Adveniente (1994), he made a strong allusion to Martyrs of Christ of various faiths: Catholics, Orthodox, Anglicans and Protestants:
37."The Church of the first millennium was born of the blood of the martyrs: "Sanguis martyrum - semen christianorum".(21) The historical events linked to the figure of Constantine the Great could never have ensured the development of the Church as it occurred during the first millennium if it had not been for the seeds sown by the martyrs and the heritage of sanctity which marked the first Christian generations. At the end of the second millennium, the Church had once again become a Church of martyrs. The persecutions of believers — priests, religious and laity — has caused a great sowing of martyrdom in different parts of the world. The witness to Christ borne even to the shedding of blood has become a common inheritance of Catholics, Orthodox, Anglicans and Protestants, as Pope Paul VI pointed out in his Homily for the Canonization of the Ugandan Martyrs.
This witness must not be forgotten. The Church of the first centuries, although facing considerable organizational difficulties, took care to write down in special martyrologies the witness of the martyrs. These martyrologies have been constantly updated through the centuries, and the register of the saints and the blessed bears the names not only of those who have shed their blood for Christ but also of teachers of the faith, missionaries, confessors, bishops, priests, virgins, married couples, widows and children.
In our own century the martyrs have returned, many of them nameless, "unknown soldiers" as it were of God's great cause. As far as possible, their witness should not be lost to the Church. As was recommended in the Consistory, the local Churches should do everything possible to ensure that the memory of those who have suffered martyrdom should be safeguarded, gathering the necessary documentation. This gesture cannot fail to have an ecumenical character and expression. Perhaps the most convincing form of ecumenism is the ecumenism of the saints and of the martyrs. The communio sanctorum speaks louder than the things which divide us. The martyrologium of the first centuries was the basis of the veneration of the Saints. By proclaiming and venerating the holiness of her sons and daughters, the Church gave supreme honour to God himself; in the martyrs she venerated Christ, who was at the origin of their martyrdom and of their holiness. In later times there developed the practice of canonization, a practice which still continues in the Catholic Church and in the Orthodox Churches. In recent years the number of canonizations and beatifications has increased. These show the vitality of the local Churches, which are much more numerous today than in the first centuries and in the first millennium. The greatest homage which all the Churches can give to Christ on the threshold of the third millennium will be to manifest the Redeemer's all-powerful presence through the fruits of faith, hope and charity present in men and women of many different tongues and races who have followed Christ in the various forms of the Christian vocation.
It will be the task of the Apostolic See, in preparation for the Year 2000, to update the martyrologies for the universal Church, paying careful attention to the holiness of those who in our own time lived fully by the truth of Christ. In particular, there is a need to foster the recognition of the heroic virtues of men and women who have lived their Christian vocation in marriage. Precisely because we are convinced of the abundant fruits of holiness in the married state, we need to find the most appropriate means for discerning them and proposing them to the whole Church as a model and encouragement for other Christian spouses."
The real crux of the canonization as this question implies, rests in the possibility of Martin Luther King Jr. being declared a Martyr for Christ, officially by the Catholic Church. Obviously he could not be neither beatified nor canonized as a martyr of the faith, since he was not of the Catholic faith. That said, the possibility of him being beatified and then canonized as "martyr of Christ" could be done, if one could find a Catholic bishop to commence the process of canonization at the diocesan level and follow through at all the other levels, just like any Catholic raised to this honor. Rome would respond accordingly.
I know of a few such cases thus presented to Rome, but have not heard of any outcomes as of yet!