Is it permissible to fulfill a vow by sacrificing an animal on a day other than the Day of Sacrifice (Eid al-Adha)? And how is the meat of the sacrificed animal to be distributed, and is it permissible to give some of it to family members?

If your verbal intention was merely to slaughter, without specifying a day like the Day of Eid al-Adha, then you may fulfill your vow on any day you wish, and you should hasten to fulfill it.

As for distributing the vowed animal, it depends on the intention of the one who made the vow. If he intended to distribute it like an animal sacrificed for Eid al-Adha, then he should do so. If he intended it for the poor and needy, then it is for them exclusively. If he did not specify a recipient and made a general vow, then its distribution is like that of charity, for the poor and needy, and he should not eat from it.