There is so much jazz music written for pipe organ it'll make your head spin.
(Just last month I played a nice toccata on O when the saints for my congregation.)
In general, at least some practitioners of most religions, and certainly of most Christian denominations, embrace almost any musical style there is.
The reason you don't hear it more often is the same reason as for many other styles: location, location, location!
Pipe organs are found almost exclusively in churches and concert halls, while jazz is more often played in clubs and bars.
music.stackexchange.com/users/3083/Kilian Foth
You are aware that the Hammond organ was created as a church organ?
It became one of the seminal instruments for Jazz, with the main reasons being portability (well, compared to a pipe organ that is) and affordability (well, again compared to a pipe organ).
The Leslie rotary speakers helped setting its sound off from congregational use.
It took a number of decades before Hammond accepted them as a legit extension of the organ: he changed the organ connectors several times in order to stop the use of Leslie speakers.
The previous compromise, namely a theater organ was still rather immobile and expensive and many were scrapped eventually when silent film died.
Since Hammond organs were designed as church organs, it is probably fair to say that price and venue were the main impediment for pipe organs as a mainstream Jazz instrument.
After all, Jazz was the music of the devil, so practice and performance in a church would have been somewhat tricky.
music.stackexchange.com/users/None/
Two Reasons that I'd suggest: Location, Context, Cost: Where are church organs?
In churches... jazz is commonly a rejected genre preference by churches that own church organs.
Remember that is today, just ponder on the rejection jazz would of received from churches in the era in which jazz originated.
Clearly the church organ wasn't off to a good start in the culture of jazz.
Many of the instruments in main stream jazz aren't that all expensive in the time of early jazz and back then it was a source of income for many families.
So there would always be a neighbour or household in the street with a piano or other affordable instruments (guitar and woodwinds).
Many of the homes would contain mini church organs too, but who wants to lunk that thing around gig to gig when you're trying to make an income?
Sound: Just like any genre,
the culture of a jazz defined its basic 'what's allowed' instruments many years ago, most of us still stick to today.
In fact, Google's definition of jazz literally contains those main stream instruments that were defined in its making.
You'll notice that all the instruments that are mentioned all share a common theme, they're mellow instruments or can be played that way.
Since the church organ is reliant on great amounts of air pressure, being mellow is simply not an option.
The church organ would simply overpower a jazz band.
Therefore, a church organ would simply throw off main stream jazz.
Overall, the reason why the church organ isn't commonly seen in jazz is because the instrument was impractical in era in which jazz was defined.
Think of rock'n'roll, it was defined by electric guitars, drums and some crazy singing, why don't we add bag-pipes to rock'n'roll?
The same goes with church organs and jazz, mainstream musical culture.
However, as technology progressed having a church organ can now be as accessible as clicking a button on your keyboard.
Since that, the organ occasionally makes an appearance in jazz, usually with the rotary effect.
See!
But still not that common because of the genre's culture.
music.stackexchange.com/users/27192/jazzboy
It does happen occasionally, but it's not common.
One example off the top of my head is Barbara Dennerlein's numerous pipe organ albums.
But it's not common for exactly the same reasons as why, for example, rock typically features guitars.
Each genre of music has its own typical instrument lineup, and while some people will experiment with other instruments most people will stick to what's standard.
music.stackexchange.com/users/28510/Micheal Johnson
"Church" organs are basically permanently installed in the building and can't be moved around to church socials in the basement (which is why the Hammond was created for the church ), or to honkytonks, juke joints or other clubs, or recording studios (which is where jazz is played).
music.stackexchange.com/users/31199/Beanluc
I'm just curious as to what about let's say a saxophone makes it very well suited to jazz music and what about the church organ makes it unsuitable for jazz music.  Is there something innate to the character of those instruments that leads them to be played in certain styles or is Jazz church organ just a style waiting to be created by some young whippersnapper?
Why don't we see jazz music played on the church organ?
music.stackexchange.com
<jazz><saxophone><organ>
music.stackexchange.com/questions/63629
music.stackexchange.com/users/7306/Neil Meyer
 "Church" organs are basically permanently installed in the building and can't be moved around to church socials in the basement (which is why the Hammond was created  for the church ), or to honkytonks, juke joints or other clubs, or recording studios (which is where jazz is played). Church organs can't be moved around to church socials or honkytonks
Church organs can't be moved around to church socials or honkytonks, juke joints or other clubs, or recording studios.