Keep in mind that what the church is asking for is to allow those members to do VOLUNTEER work for the church.
How is that any different from the "Americans don't want to pick cabbage" argument?
Except, of course, that illegal cabbage pickers earn 10 cents an hour and these volunteers wouldn't?
The difference is that there is no money motive at all in volunteer work. There are no paid ministers or missionaries so nobody is being potentially deprived of any income.
In many cases these persons wanting to server missions are in the USA illegally because their parents snuck them in years ago when they were kids. I fail to see how anyone is harmed by this.