Is the husband considered to have fulfilled his vow by slaughtering the sheep if his wife slaughtered it on his behalf due to his inability to do so?

The validity of a vow depends on the intention of the one who made it. If the husband intended to slaughter the animal himself, then his wife's slaughtering would not be sufficient. However, if he intended merely for the slaughtering to occur, regardless of who performed it, then his wife's slaughtering would be sufficient on his behalf.