"Is the group obligated to pay the blood money for an accident that led to the death of two men, even though the person who caused the accident had previously refused to participate in supporting the group?"

The blood money for an accidental car crash is borne by the 'asabah (agnatic relatives), because the Prophet, peace and blessings be upon him, ruled that the blood money is to be paid by the 'aqilah (male agnatic relatives). Therefore, if you are among the 'asabah of the young man and there isn't sufficient (funds) from the closer relatives, then you must contribute to the blood money. His lack of contribution to other matters that are not religiously obligatory does not justify your refusal.