All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His Slave and Messenger.
When one makes a vow to Allaah, the Exalted, to do a certain act of obedience to Him, he has to fulfill his vow and do that act. Whoever fails to fulfill his vow should offer expiation like for breaking an oath. Hence, it is incumbent on your mother to perform ‘Umrah if she is physically able to do so. If she is unable to perform it due to an illness that is not chronic, then she should wait until she recovers from her illness and perform it when she gets better. However, if her illness is chronic, then she can delegate someone to perform ‘Umrah on her behalf. It is mandatory on the person to delegate someone to perform the obligatory Hajj on his behalf if he is physically unable to perform it. The same applies to a person who makes a vow to perform ‘Umrah but is physically unable to perform it himself. If you perform it on behalf of your mother, then it will be sufficient for clearing her liability (and her vow will be fulfilled). If she is financially unable to delegate someone to perform ‘Umrah on her behalf and cannot find someone to perform it on her behalf for free, then it is incumbent on her to offer expiation like for breaking an oath.
Allaah Knows best.