I was about 2 years away from serving a mission. Looking at things, I don't think i would have been able to, considering that you have to travel and might have to drive a car as a missionary, a valid U.S. ID is essential. It's difficult to be honest, but I would say that if he doesn't want to expose himself, he needs to just say he doesn't want to serve a mission, or he needs to tell the Bishop exactly what is going on with his status and maybe tell him to ask what the Church's policy is on having undocumented people serve missions. I wouldn't have him sign up and ready to leave and then have someone ask for his ID or fall in a situation were he needs to show an ID and have him go through the embarrassment then.

If he decides to be honest about his status, I've also heard of people being able to serve their missions locally in their city, or a nearby city.