Sometimes google searches are a bit of a pain...but if you keep trying, you eventually find what you need.
The Mormon church arranged for a Utah senator to write a law to shield churches from prosecution for knowingly allowing illegal aliens to be ministers or do volunteer missionary work for them. Kim Farah, a spokeswoman for the Church of Jesus Christ of Latter-day Saints, released a statement saying the church asked Sen Robert F. Bennett, Utah Republican, to sponsor the provision, which she called a "narrow exception to the immigration act."
U.S. Rep. Tom Tancredo (R-CO), today criticized a loophole in U.S. immigration law that was quietly created by Sen. Robert Bennett (R-UT) in 2005. The "Bennett Loophole" created a broad exemption that allows religious organizations to knowingly transport, aid, abet and harbor illegal aliens....The Senator claims he authorized the far-reaching provision to assist the Mormon church in utilizing illegal alien missionaries and volunteers.
A Salt Lake Tribune link I found unfortunately no longer worked.
So yes, apparently, the church wants to give undocumented immigrants, raised in the US, the chance to serve inside the US without the church getting punished for it; and apparently the church has been successful in convincing Congress to pass a bill that allows the church to do that. Unless anyone interprets these articles differently than I do...or unless the articles are false.
I'd need the leadership handbook to be certain, but it certainly seems like the church doesn't discriminate based on citizenship when sending out missionaries.