Leading Edge is BYU's Sci-Fi/Fantasy magazine. It accepts admissions from anyone and pays for the stories it accepts. Sometimes these are student submissions, but more often than not, the submissions come from the world at large. The support from faculty for the magazine has been variable over the years. Over the past few years, it stopped getting the support from the English faculty it needed and moved over to another department. This was mainly because the professors who had been more supportive died or retired. The remaining professors didn't think it was "literary" enough, or something of that sort. I don't know the exact details.
There are a lot of LDS Sci-Fi/Fantasy authors, and many of them went to BYU. Brandon even teaches a Sci-Fi/Fantasy creative writing class there. When I was at BYU, Dave Wolverton taught that class, and there was a Sci-Fi/Fantasy literature class offered too. I think support for that sort of thing has depended more on the professors who are there and students willing to put in the time and effort to make it happen.
You can read more about it at:
http://www.leadingedgemagazine.com/