Reference: Majmoo’ Fataawa wa Rasaa.il ‘Abdul-‘Azeez ibn Baaz – Volume 16, Page 407
Question: [Regarding] the ‘umrah, is it permissible to do so on behalf of another if the person is unable [to do so himself] due to an illness?
Response: The ‘umrah is similar to the hajj. If the person is incapacitated due to an illness that is not anticipated to pass, or due to old age, then he can appoint someone to perform the ‘umrah on his behalf, [just] like the hajj.