Is it permissible to cook the two sheep that have been vowed to feed those breaking their fast in the mosque, and is it true that a vow is exclusively for the poor and it is not permissible for family members to eat from it?

The vow mentioned is a "vow of exchange" (nadhir mu'awadah) which is prohibited. However, it must be fulfilled. The one who made the vow is not allowed to eat from it unless they intended or stipulated that. The meat of the vow must be distributed to the poor and needy. If the one who made the vow specified a particular recipient, they must adhere to that. The two sheep should be of the type that is permissible for sacrifice (udhiyah).