Habitat for Humanity may also be an idea.
You could also contact Home Depot, a local contractor, or the local news station. You may be able to drum up enough volunteers to have an old fashioned wall raising party.
My dad added on to our house when we were kids. When he had the foundation ready, all the neighbors gathered and they had a wall raising. He had all the supplies there and each person brough a hammer. They had all the bottom wall built and up before lunch. After lunch, they put in the ceiling/floor. It was done very quickly. My dad was then able to work on the structure of the downstairs. When it was time to wall up the top, same thing, we had another wall raising. Mom and us girls cooked and were the hostesses. Oh, I think we did the same thing for the roof. He shingled with only 1 or 2 other people.
Of course, it was funny because there was one neighbor who did not come. He just stood on the other side of the fence watching and telling everyone that they were doing it wrong.
Well, 2 decades and 2 hurricanes (Andrew and Katrina) later, it is still standing.
Anyway, back on topic here. A lot of times, an event such as this can get a lot of community support. If you can get a construction company to volunteer to do the foundation, then gather a bunch of volunteers to do the rest. All you need is a couple of professionals to supervise.
Only problem I see is that if the land they are on now is where the new house would go, is where will they stay while it is being built. Extreme Home Makeover does it in a week, but they have hundreds of volunteers. I also don't think that EHM would move on this story. They usually help families with a little bit of tragedy or major health concerns, e.g. mom died and left dad with 8 kids, daughter has major heart problems and must live in a sterile environment, son was shot and left paralyzed, etc. It won't hurt to ask though.
I would contact Oprah, Martha, the Today Show, etc. Write letters to all of them. It will probably be a numbers game. The more people you contact, the bigger chance of someone responding.
Oh, just thought of something. The mobile could be moved to a rental mobile community while it is being built. Of course moving it can be thousands of dollars depending on how far. You may be able to get a towing company to volunteer for that as well. If you get the news involved, more companies will volunteer for the free publicity.
OK, overload reached. Good Luck!!!