For amusement's sake, let's take six choral accompaniments I am currently preparing and arrange them from most to least idiomatic for the organ:
How dazzling fair (Charles Wood)
I was glad (C. H. H. Parry)
Zadok the Priest (G. F. Handel)
O how amiable (Ralph Vaughan Williams)
Wie lieblich sind deine Wohnungen (Johannes Brahms - from Ein Deutsches Requiem)
The Hour has Come (Srul Irving Glick)
Most people, even other musicians, are probably not aware of the extent to which the organist-accompanist creates his own score for the works he performs. A part like "How dazzling fair" can be played as written and sound fairly good - the Glick, on the other hand, is literally unplayable as it appears in the score and requires extensive modification. As a collaborative pianist, your role is comparatively simple - to play the score accurately and musically, while being sensitive to the needs of the soloist or ensemble that you're accompanying. (I still do quite a lot of collaborative piano work, and can vouch for the fact that this is no easy task!) Yet accompanying on the organ adds another layer of complexity entirely:
1. If your part is written for the organ and has particularly detailed instructions, you must select stops and distribute the part across the various manuals in a way which suits the instrument, choir and acoustic. Because you will have limited rehearsal time, you must be able to change any aspect of this plan at a moment's notice.
2. If your part is written for the organ but lacks detailed instructions, you must develop a registration scheme entirely from scratch, trying to find appropriate colours to suit the style of the piece and looking for opportunities to "solo out" an important inner voice, plus everything listed under #1.
3. If your part is written for piano (or for "piano or organ", which means the same thing), you must edit the part to eliminate pianistic figuration, generally consolidating arpeggiated or Alberti-bass figuration into sustained chords. A pedal line can typically be generated by playing the bass voice of the piano part in the pedals, but this doesn't always work and it will often be necessary to compose a pedal line based on the implied bass line of the part. Generally, this will result in a rather "thin"-sounding part which must be filled out by adding additional notes from within the chords. Plus, of course, everything listed under #2.
4: If your part is an orchestral reduction, it is likely intended for performance on piano and thus has all the problems associated with group 3. In addition, it will probably be necessary to consult a recording or score of the orchestral original in order to select appropriate registrations. Placing the bass line (double bass, cello, or tuba parts) in the pedals makes it possible to write in an inner voice (cello, bassoon, viola, or horn) to be played by the left hand, often using a solo registration.
It's a tough life, in other words. Not all of these issues are unique to the organ - a good collaborative pianist will know the orchestral score for a concerto reduction, and add things to the part accordingly - but in combination, these problems become somewhat daunting. Unfortunately, the Glick is about as bad as it gets (group 4+++) and so will require almost total rewriting before I'm satisfied with it.
On the plus side, organ accompaniments don't get much more fun than "Zadok" or "I was glad".