Is it permissible for me to perform Hajj on behalf of my mother, who is unable to perform the Hajj pilgrimage due to her old age and illness, while I intend to perform Hajj for myself this year?

If the mother's condition is as mentioned in the question, then there is no harm in performing Hajj on her behalf, as there are narrations that permit it. However, the permissibility of performing Hajj on her behalf is subject to conditions mentioned in Fatwa no. 43931.