I spent considerable time trying to track down information about this topic...and then the Salt Lake Tribune goes and publishes an article that would've answered a lot of my questions.
This news from Cincinnati, of all places (and I wonder if the missionaries in our ward know more about this--I'll have to ask them).
For those who still doubt what the church's take is on having undocumented immigrants serve missions in the US, here's an article, along with ample quotes from Elder Holland, on the issue.
It seems clear that, unlike many conservatives, church leaders do not view illegal immigration as much of a sin (although they do discourage it).
I support the church 100% on this issue.