First of all, we would like to state that it is originally permissible to appoint someone to do something on behalf of another in relation to acts of worship when one has some legal excuse, such as illness and as such.
As regards authorizing someone to throw the pebbles on behalf of another in Hajj, Dr. Husam ad-Din Ibn Musa `Afana, professor of Principles of Islamic Jurisprudence at Al-Quds University, Palestine, states: In the first place, a pilgrim must perform all rituals of Hajj by himself. However, Muslim scholars legalize authorization in throwing the pebbles and slaughtering the hadi.
It is permissible to throw the pebbles on behalf of another person who has some legal excuse, such as sickness, pregnancy for women, old age, or weakness. Also, it is permissible for a woman to authorize someone to throw the pebbles on her behalf if she cannot bear overcrowding of the pilgrims and would be harmed if she did. In this case, the authorized person must throw his pebbles at first, then he can throw the woman’s pebbles on her behalf.
It is also permissible to authorize somebody to slaughter the hadi on one’s behalf in Hajj, whether there is a legal excuse or not, according to my knowledge. However, it is better for the pilgrim himself to undertake slaughtering, following in the footsteps of the Prophet (peace and blessings be upon him), who slaughtered63 cows then authorized `Ali ibn Abi Talib (may Allah be pleased with him) to slaughter the rest. (Muslim.)
As for the rest of the Hajj rituals, i.e., Tawaf (circumambulation of the Ka`bah), Sa`i (going between Safa and Marwah), staying at Mina and Arafat, it is not permissible to perform them on behalf of another. Such are the main rituals and obligatory acts of Hajj that a pilgrim must perform himself.