Jackman Music holds a virtual monopoly on "paid" sheet music. Chances are, if you sang a choir piece or bought a piano arrangement of a Mormon hymn, it was published by Jackman. They specialize in music that is appropriate for singing in church settings (sacrament meetings, conferences, and so on). They are what you would call a "traditional" publisher.
Believe it or not, Deseret Book does NOT publish sheet music. They sell it in their stores, but they seem to be happy enough to let others do the publishing. Deseret Book does use an imprint, Shadow Mountain Records for recorded music, but I'll explore that later.
There are a few "paid" publishers that sell music online. This new model allows one to download a pdf file, print as many copies as needed, and purchase labels to affix on each copy to make each one legal. Since there is no printing on their side (except for the labels), they're able to sell at prices lower than "traditional" publishers. However, your choir will have to be ready to shuffle loose pieces of paper.
LDS Music Source, famous for their green labels, seems to be the oldest. There's a good chance you may have sung one of their pieces. They feature the music of David A. Zabriskie, and a couple of other composers.
HolySheetMusic, using gray and red labels, is somewhat newer, run by none other than David A. Zabriskie. They seem to offer a larger collection of music from composers like Daniel Carter, Kathleen Holyoak, and others. Plus, they claim that the copyright remains with the composer.
MusicSpoke is another newer experiment featuring allowing anyone to download pdf files for a price.
Then come a whole slew of free publishers. The most famous one is Free LDS Sheet Music. Practically anyone can post their music and have it available for free. If all you wish is to expose the world to your music without earning cash, there is no longer any excuse. Check out this site, and you will find your outlet. However, like the last three websites, this is download only.
To summarize all of the above, there exists only one prominent publisher that offers printed Mormon sheet music. So, if you're an aspiring Mormon composer who hopes to hold a good-looking physical booklet of your music in your hands, Jackman Music is really your only option.
That is, of course, unless you do it yourself. Many Mormon composers have gone this route. One is my former BYU colleague, Cameron Rose. He created his own publishing company, Providence Music, for the main purpose of promoting and making available his music. As I contacted him and asked about his business, he was happy to tell me how it all works, and gave me valuable advice. In a nutshell, the technology exists for anyone to do this on their own, if you're willing to do the extra work.
Another is my friend at church, Geoff Groberg, who is also currently helping me put my business together. He specializes in folk music, produces sound recordings, and offers sheet music for free.
And that's where my new upcoming publishing company fits in. At first, it will only sell my music, targeting Mormon/Christian audiences with one imprint, and more classical audiences with the other imprint. Later, I'd revisit to see if it's able to print other composer's works as well.
My competition will be the ever-pervasive Jackman Music, the online sticker download market, and the flood of free downloadable music on the other side. With this backdrop, I will have to sell my music as being something new and worth buying.
I'll let you know how it goes as things move forward. Perhaps you may be inspired to take a similar path.