Any organ in a church would be a "church organ" right? Unless you specifically mean a pipe organ.
Of course organs are used in jazz: Jimmy Smith for example. Wild Bill Davis is another.
Do the old giant theater organs not count? Certainly jazz was played on those organs.
If you mean something like "why isn't jazz played on organs in a church", the obvious answer is churches already have music for services and it isn't jazz.
Now "what is jazz?" needs to be unpacked.
If you consider certain blues or gospel styles to be essentially forms of jazz, but with different lyrical content, then you have jazz in church. Then question the circles back around to: does the church have an organ?
I'm reminded of a Pentecostal family I was friends with in high school who ran a small church. The eldest song played jazzy gospel keyboard. They had organs in the house and in the church. There wasn't much difference in his playing in the church or in the house, although his playing was probably funkier in the house and more subdued in church. His playing was totally ad lib. I think it fair to say his style was jazz.
Just like European "classical" music had a secular/sacred dichotomy, so does African-American music. I read a very good book that discusses the topic: Murray, Stomping the Blues. The chapter The Blue Devils and the Holy Ghost is the main discussion. Don't dismiss the "blues" in the title, the book covers Ellington, Basie, Armstrong, etc. The book broadly makes the point that the only difference between secular/sacred, jazz/gospel, is the venue.
I think the answer is if you go to a church with the right musical tradition, and that church happens to have an organ, you will hear jazz on a church organ.
There may not be a lot of churches like that compared to other churches, or they may not be in your social circle.