If my father-in-law passes away without a Will and his disabled son currently lives at the house, what

are his rights?

If there is no will, the property passes by intestate succession--the rules for who gets what when there is no will. Since he is your father-in-law, he presumably has at least two children: the disabled son, and your spouse. That means that his children will share in his estate, or everything he owns--e.g. if there are two children, the disabled son and your spouse will end up jointly inheriting the house and will have to either between them figure out what to do with it, or else sell the home and split the proceeds. (This answer assumes your father-in-law is not married when he passes; if he is, the spouse will have a claim to the home, too.)
If your father-in-law wants to make sure his disabled son can live there, what he should do is one of the following:
1) Leave the son the house, and leave his other child(ren) everything else, assuming that he has enough other assets to more or less equal the house's value, so that everyone is treated approximately fairly (assuming he wants to be fair); or
2) If his other child(ren) can take care of him/her/themselves, leave the house and money for its upkeep to the disabled son and not worry about the other children; or
3) He can give the disabled son a "life estate" to live there for the rest of his life (or until he voluntarily chooses to move out) while giving the "remainder" interest to his other children (or their children/his grandchildren), so they inherit when the disabled son passes away or leaves.
There are some other variations on the above, too, that he can do, but all would need to be in a will. A will does not need need to be too expensive: for example, I'm a NJ lawyer and do wills in NJ for $500 (including also getting a power of attorney and "living will" for medical needs). You can help your father-in-law find an affordable attorney.