First of all, I believe that this skill can be learned like any other skill we have to do. Some people believe that only geniuses can play anything at sight. We all have read stories about Bach's notorious ability to play new music with ease. However, we tend to forget how much Bach and other geniuses had to practice in order to achieve such heights. In fact, Bach himself admitted once that it is not possible to sight-read everything.
Ideally a great musician should be able to sight-read compositions of medium level of difficulty with ease and fluency. To achieve this, we need to practice specifically sight-reading. It is a good idea to devote some 15 minutes of each practice session just for that.
I think many people like playing organ pieces at sight. It gives them pleasure to hear new music every day. However, the success of sight-reading exercise depends on the playing speed, difficulty level of the composition, and organist's attention to detail. In fact, that's where many people fail.
Ideally, we should feel no tension and stress when sight-reading. If we do, this means either our practice tempo is too fast or the difficulty level of the piece is too great or both. It is probably better at the beginning to play easy music at sight in a very slow tempo with no mistakes than to play it up to concert tempo but constantly failing in each measure.
If sight-reading all parts together is too advanced, you can try playing separate parts and voice combinations. By the way, this is exactly the same method I use to learn new clefs. From there we can go into practicing music in open score notation and orchestral reductions.
Find a collection of organ music of your level that you love and start sight-reading one page a day. If you practice according to the above suggestions, keeping in mind slow tempo, difficulty level, attention to detail, solo voices, and voice combinations, you will be surprised how much stronger your skill in playing new music will become. You just need to be very systematic about it and stick with it at least for several months.
If you really want to develop unbeatable sight-reading skills, check out my systematic Organ Sight-Reading Master Course. To complete the practice material of this course will only take 15 minutes a day of practice but you will learn to sight-read any piece of organ music effortlessly.