Is it permissible to offer the vowed month's alms in a lump sum or in installments, and is it permissible to give a portion of it to relatives who are well-off?

The phrase "الأقربون أولى بالمعروف" (Closest relatives have the first right to kindness/favor) is neither a Hadith nor a Quranic verse. It is permissible to disburse the vowed amount for the sake of Allah in installments, as the intention specifies the general wording in vows. It is not permissible to give the vowed money to parents or to siblings whose maintenance is legally obligatory upon you. However, it is permissible to give it to siblings whose maintenance is not obligatory upon you. If by "for the sake of Allah" you intended charity, then it is not permissible to give it with the intention of a gift, as charity differs from a gift.