Is it permissible for my mother to wear shoes during the Sa'y to help her continue, given that she suffers from "calcaneal spur" which causes her pain when standing for extended periods?

It is permissible for your mother to wear shoes during circumambulation (Tawaf), the ritual walk (Sa'i), and all Hajj rites, provided she ensures they are free from impurities before entering the mosque. If it is permissible to pray with pure shoes, then performing Tawaf and Sa'i with them is even more permissible.