Is the wife obliged to do anything because she authorized her son to stone the Jamarat due to her illness, and she left the farewell circumambulation due to time constraints? And is her Hajj valid?

The wife's Hajj is valid if she appoints her son or someone else to perform the Rami (stoning) on her behalf due to an excuse that prevents her from doing it, such as illness or extreme crowding. Otherwise, a dam (a sheep to be slaughtered in the Haram and distributed among its poor) is obligatory upon her. As for the Tawaf, delegation is not permissible for it. It is more cautious to slaughter a sheep in the Haram and distribute it among its poor.