Is it permissible to pray on a chair for someone who feels pain in their feet and is unable to prostrate regularly, and is the prayer of one who sits on a chair only for prostration valid?

If prostrating and sitting for the Tashahhud cause severe anxiety and remove khushūʿ (devotional humility) due to pain, then this is a hardship that waives prostration on the feet. In such a case, it is permissible to sit on a chair for prostration, for sitting between the two prostrations, and for the Tashahhud, and to gesture for bowing and prostrating. If your relative's condition is similar to yours, then what you are doing is correct and your prayer is sufficient.