Is it permissible to perform Hajj on behalf of my mother while she is still alive, if she is unable to ride in a car due to an illness that causes her to faint and vomit, and the distance to Makkah is 450 km, knowing that I have already performed the obligatory Hajj for myself?

For the obligation of deputizing someone for Hajj, it is stipulated that the person on whose behalf Hajj is performed must be unable to perform Hajj themselves, and they must possess wealth.

If the mother is able to perform Hajj herself after treatment that prevents fainting and vomiting, or by flying, then she is obligated to perform Hajj herself. If she is unable to do all of that, then it is obligatory to deputize someone to perform Hajj on her behalf if she possesses wealth, based on the hadith of the Khath'amiyah woman.

It is also stipulated that the deputy must have first performed Hajj for themselves, based on the hadith of Shubrumah.