Performing Umrah on behalf of a deceased person is permissible according to the consensus of scholars. The reward of the Umrah reaches the deceased by Allah's permission. This ruling is derived by analogy (qiyas) from the explicit permission for proxy Hajj for the deceased, and from the general hadith that permits good deeds performed on behalf of the dead, such as charity and supplication.
Sheikh Ibn Baz stated that performing Umrah on behalf of a deceased parent, relative, or any Muslim is a righteous act and a form of birr (kindness) that benefits the deceased. The person performing the proxy Umrah should have already completed their own obligatory Umrah (for those who consider Umrah obligatory). They should make the niyyah (intention) at the miqat on behalf of the deceased: 'Labbayk Allahumma bi-Umrah on behalf of [name].'
Sheikh Al-Albani confirmed the permissibility based on the hadith where the Prophet permitted proxy Hajj for the deceased, and Umrah follows the same ruling. He also noted that the person performing the proxy Umrah earns their own reward for the good deed, while the full reward of the Umrah goes to the deceased. There is no limit to how many proxy Umrahs can be performed for the same deceased person.