Reference: Majmoo’ Fataawa wa Rasaa.il ‘Abdul-‘Azeez ibn Baaz – Volume 16, Page 406
Question: Some of the scholars say that the Prophet (sal-Allaahu ’alayhi wa sallam) granted permission for the hajj to be performed on behalf of the deceased – without [specifically mentioning] the ‘umrah. So is [the performance of] the ‘umrah [on behalf of the deceased] correct? We are in need of knowing the evidences, so provide us [with them], and may Allaah reward you with good.
Response: It is permissible to perform ‘umrah on behalf of another, just like the hajj – whether the beneficiary is deceased or unable [to do so] due to old age or an illness which is not anticipated to pass; so hajj can be performed [on his behalf], and the ‘umrah [too].
A man came to the Prophet (sal-Allaahu ’alayhi wa sallam), and said: “O Messenger of Allaah, my father is very old, he cannot perform hajj [himself], nor can he ride on a mount; should I perform hajj on his behalf and ‘umrah [too]?” [So] he said [to him]:
«Perform hajj on behalf of your father, and ‘umrah [too].» – transmitted by Imaam Ahmad no.15751 and an-Nasaa.ee no.2621.
So there is no harm in performing hajj [and ‘umrah] on his behalf.