Who gave the Reformers the Authority?

"...Who gave the reformers the authority? It is my understanding that neither Luther nor Calvin were even Bishops. So what authority did they have to bring schism to the Body?"[Source]

One unfortunate problem with this question is that Protestants do not argue the church does not have any authority. The church always has authority. What Protestants argue is the church does not have infallible authority. As David King points out in his book Holy Scripture: The Ground and Pillar of Our Faith,

"What sola scriptura presupposes is that there is no need for the magisterium of the Roman communion to define dogma infallibly. We readily submit to duly constituted authority in Christ's Church, but we reject an institution that has made void the commandments of God by the traditions of men" [p.259].

Luther was given authority by the church of his day:

"But perhaps you will say to me, 'Why do you, by your books, teach throughout the world, when you are only preacher in Wittenberg?' I answer: I have never wanted to do it and do not want to do it now. I was forced and driven into this position in the first place, when I had to become Doctor of Holy Scripture against my will. Then, as a Doctor in a general free university, I began, at the command of pope and emperor, to do what such a doctor is sworn to do, expounding the Scriptures for all the world and teaching everybody. Once in this position, I have had to stay in it, and I cannot give it up or leave it yet with a good conscience, even though both pope and emperor were to put me under the ban for not doing so. For what I began as a Doctor, made and called at their command, I must truly confess to the end of my life. I cannot keep silent or cease to teach, though I would like to do so and am weary and unhappy because of the great and unendurable ingratitude of the people. And even if I were not a Doctor, I am, nevertheless, a regularly called preacher and may teach my own people with writings. If others have desired these writings of mine and have asked for them, it is my duty to accede to their request. For I have never pushed myself in or desired or asked that anyone should read these writings, but have acted just like other pious pastors and preachers. They write books and neither prevent people from reading them nor drive them to do so; thus they teach throughout the world. They do not run and sneak like these worthless, uncalled knaves into the offices of others without the knowledge and consent of the pastors; but they have a definite office and position by which they are driven and compelled." [LW 13:66]

The problem is the Roman church at Trent anathematized the Gospel. The Roman authority has never been infallible. By its proclamations she denied the Gospel, turning her into a sect.

The Church gave Luther authority, but this Church authority did not submit to the ultimate authority: the sacred scriptures. The Reformers had no choice but to point this out, and forge ahead as the church always does.