I am a former lds missionary. I served in the east coast. There is nothing wrong with being undocumented and being a missionary, especially in our situation. I told my bishop exactly what my situation was and he understood perfectly well. My brother did the same also. Furthermore, the general authorities of the church were fully aware also. The church will make accomodations in regards to the method of travel. They will also make sure the mission call is inside the states (unless there's an extreme need of him outside the country, which is highly unlikely).

Make sure the bishop knows. The worst thing he can do is turn in his mission papers without the bishop knowing about it. Is it a spanish ward or an english ward? It worked for me, my brother, and many of my friends. It was an amazing experience. Let me know if you have more questions.