He got sick on the day of Eid; should he delegate someone to stone the Jamrah on his behalf, or should he delay stoning the Jamaraat until the last of the days of at-tashreeq?

If some impediment arises that prevents a person from stoning the Jamrah on the day of Eid, then he may delay it until he is strong enough to do that, because the Prophet (blessings and peace of Allah be upon him) gave permission to the herdsmen to stone the Jamaraat on one day and to refrain from doing so on one day, and he did not tell them to appoint someone to do that on their behalf.

It is not permissible to be careless about the matter of appointing someone to stone the Jamaraat on one’s behalf, because stoning the Jamaraat is one of the rituals of Hajj. Allah, may He be exalted, says (interpretation of the meaning):

“And complete the hajj and ‘umrah for Allah”

[al-Baqarah 2:196].

So it is not permissible to appoint someone else to do that just because one feels tired, or because it is crowded. We say: As for feeling tired, if a pilgrim is tired all the time, as in the case of a pregnant woman, old man or old woman, then in this case he or she may appoint someone else to stone the Jamaraat on his or her behalf. But if the pilgrim is affected by mild illness that he hopes to recover from by the end of the days of at-tashreeq, then it is not permissible for him to appoint someone to do that on his behalf. End quote.

Majmoo‘ Fataawa Ibn ‘Uthaymeen (23/115)