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