Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is permissible to offer udhiyah on behalf of deceased relatives as an act of charity. And the latter is expected to reap the rewards for the same.
In responding to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Originally, sacrifice (udhiyah)– as is the case with other rituals — is prescribed on the living. Nevertheless, it is permissible for a person to offer udhiyah on behalf of his deceased relatives as an act of charity. And the latter is expected to reap the rewards for the same.
We learn from the Sunnah the Prophet (peace be upon him) allowed people to do hajj, to fast and offer charities on behalf of the deceased and praying to Allah to give the rewards.
That is like offering gifts to them even as we do it to the living. In the case of the dead, however, unlike the latter, we instead of material gifts we give only the spiritual gifts.
Some people make the mistake of dismissing such a concept as absurd. They are wrong since such matters belong to the realm of ghayb (unseen world); since the Prophet, has approved them, we cannot find fault with it.
We can infer the same conclusion from a report from Aishah, the beloved wife of the Prophet (peace be upon him). She said,
“When the Prophet (peace be upon him) offered sacrifice, he prayed thus: I sacrifice in the name of Allah. O Allah, accept this sacrifice on behalf of Muhammad, his family and the ummah of Muhammad – saying this he sacrificed the sheep.”
The Ummah of Muhammad includes the living as well as those who have passed away.
Almighty Allah knows best.