It is possible but however it's not easy, you have to legally adopt the child which involves a home study by a state approved adoption agency and a petition filed with USCIS to classify that child as a orphan, which means your cousin have to give up parental rights in writing to classify him as a orphan, that is provided if the absent parent has also abandoned the child.
Please note if you do all that, that child will still not be able to sponsor his real parents/siblings ever for any immigration benefit.
