I was fortunate to have studied with a teacher from the French school - himself a student of Marcel Dupre. Improvisation was part of my lesson plan. I was drilled in playing chord progressions in all keys and in getting from one key to another on demand. I was also taught to fill out harmonies in the French style. This was a part of every organ lesson for many years.
My teacher had also worked as a theater organist, helping to pay for his lessons while he was in Paris in the 30s. Clearly theater organists improvise a great deal, so classical organ improvisation always seemed natural to me.
When I used to play for a church, I would sometimes improvise preludes or postludes. In some cases, I would use a hymn tune that was being sung in the service as a theme. In most of these cases, I would run through possible things I might do in advance at home - but the real-time performance was always spontaneous.
In both church and synagogue work, I would often improvise short transitional pieces or meditations on demand. There is no way I could have enough sheet music with me (and find it fast enough) to get through such situations, so this method is extremely practical.
Many of the big organ works we perform also began life as improvisations.
Because the organ is most often played by itself (I am excluding choral pieces here) it is an ideal platform for improvisation.
Les