If you are asking specifically about The Stations of The Cross as we know them today in the West, that is to say: the practice of placing symbolic artwork around a worship space depicting the events of Jesus' final hours and the practice of moving about from station to station in prayer, you can probably pin it on The Franciscans (and by extension St. Francis of Assissi) sometime near the end of the 17th century in central Europe.
Taken in this regard The Stations are a symbolic representation of a physical, bodily pilgrimage to the purported locales of said events, which may have begun within a generation or two of Christ's crucifixion.
Tradition asserts that the Blessed Virgin used to visit daily the
scenes of Christ's Passion and St. Jerome (c. 347 – 420) speaks of the crowds of
pilgrims from all countries who used to visit the holy places in his
day. (The Catholic Encyclopedia - 1907)
Fast forward to The Middle Ages and The Crusades. (The military campaigns enjoined by The Catholic Church to re-establish Christian access to said holy places.)
In 1342, Pope Clement VI, by the Papal bulls Gratiam agimus and Nuper
charissimae declared the Franciscans as the official custodians of the
Holy Places in the name of the Catholic Church. (The Bull of Clement VI - 1342)
Then, in central Europe during the 15th and 16th centuries, Franciscans started to create outdoor sculptures and shrines that would replicate these holy places of Jerusalem. These were usually placed, often in small buildings, along the approach to a church, as in a set of 1490 by Adam Kraft, leading to the Johanneskirche in Nuremberg. (1)
But to get to what - I think - is the crux of your question:
Realizing that few persons, comparatively, were able to gain these by
means of a personal pilgrimage to the Holy Land, Innocent XI, in 1686,
granted to the Franciscans, in answer to their petition, the right to
erect the Stations in all their churches, and declared that all the
indulgences that had ever been given for devoutly visiting the actual
scenes of Christ's Passion, could thenceforth be gained by Franciscans
and all others affiliated to their order if they made the Way of the
Cross in their own churches in the accustomed manner.
Innocent XII
confirmed the privilege in 1694 and Benedict XIII in 1726 extended it
to all the faithful. In 1731 Clement XII still further extended it by
permitting the indulgenced Stations to all churches, provided that
they were erected by a Franciscan father with the sanction of the
ordinary.
At the same time he definitely fixed the number of Stations
at fourteen.
(The Catholic Encyclopedia - 1907)
(1) Schiller, Gertrud, Iconography of Christian Art, Vol. II, p. 82, 1972 (English trans from German), Lund Humphries, London, ISBN 0-85331-324-5
