Is it permissible to give votive money to family members, such as one's mother or sisters, or is it exclusively designated for the poor and needy? And is it permissible to give it in installments?

If the one making the vow does not specify a destination for his vow, it should be spent in the same channels as Zakat. There is no objection to paying the vowed amount in installments, over two payments, if no specific time for payment was set, because fulfilling a vow is not an immediate obligation.