Would blood be incumbent upon my mother if she authorized me to stone Jamrat al-Aqaba al-Kubra on her behalf due to fear of congestion, knowing that I stoned on her behalf while she was close to the Jamarat?

If a mother is unable to cast the pebbles due to illness or severe overcrowding, it is permissible for her to appoint someone to cast them on her behalf, and there is no blame upon her. Similarly, those with valid excuses who find it difficult to cast the pebbles during the crowded times are permitted to depart from Muzdalifah and cast them before dawn.