Is it permissible to slaughter one sheep with the intention of fulfilling two separate vows (one from the husband and one from the wife) for the recovery of their daughter, and is it permissible to eat from the meat of this vow?

Each of you must slaughter a sheep because each vow is independent. It is not permissible to eat from a vow of requital (nadhr (definition) al-mujazah) that has been designated for the poor. If any of it is eaten, its equivalent value must be guaranteed to the poor, either by the value of what was eaten, or by an equivalent amount of meat, or by purchasing a share in an animal like it and participating in its slaughter.