The Marine Corps doesn’t have chaplains. But that’s because the Marine Corps is just another corps of the Navy, and so Navy chaplains minister to Marines. It would be like saying that the Navy Corps of Engineers doesn’t have chaplains either, which is also technically true, but misleading.
One would assume that Starfleet, the pseudomilitary organization of the United Federation of Planets (which Earth, it is hoped, will be a founding world of) won’t have chaplains either, but for entirely different reasons than why the Marine Corps and the Navy Corps of Engineers don’t have chaplains.
Star Trek creator Gene Roddenberry was an atheist, and so he imagined a future for Earth without war, without money, without poverty, and… without religion.
There are fleeting references to religion in Star Trek (now known by the retronym “The Original Series”), but these can be explained away as turns of phrase.
By the time of Star Trek: The Next Generation, the writers were much clearer on Roddenberry’s idea that humans have outgrown the need for religion (see for example “Who Watches the Watchers?”).
On Star Trek: Enterprise, which is set well before the original series, Dr. Phlox (John Billingsley) mentions exploring Earth’s religions, attending services at the Vatican and elsewhere on the planet. But I don’t recall any mention of the Denobulan doctor’s own religion from his home planet. Presumably he doesn’t have one.
So by the time of Captain Kirk, it seems no human practices any religion. However, Star Trek commentator Steve Shives posits that Starfleet will still need chaplains. To some extent.
This 17-minute video is hilariously blasphemous, in my opinion. Others might say it’s inappropriately sacrilegious.
The open thread question: Assuming human civilization continues to exist to the 24th Century, what part, if any, will religion play in people’s lives?