Is it permissible to benefit from things that are most likely stolen, given that not using them would result in significant problems?

If you are not certain, or you have a strong suspicion, that purchased items are stolen, then there is no harm in benefiting from them, as the original presumption is that the seller owns what he sells. However, if you know that they are stolen, then it is not permissible for you to benefit from them, and you must explain to your family the prohibition of buying stolen goods and the obligation to return them to their owner.