All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.

Dear brother in Islam, thank you for your question, which emanates from a God-fearing heart, since it shows your concern for the affairs of your fellow Muslims. Also, we commend your pursuit of Islamic counseling.

As regards your question, the basic ruling is that it is not permissible to perform tawaf on behalf of another person who is attending the Hajj. This is because even if a Muslim performing Hajj may be unable to perform tawaf around the Ka`bah, he or she may be carried by anyone to perform tawaf. So, one has to perform tawaf  for himself or herself even if he or she is carried on the shoulders of another person.

Responding to your question, Wael  Shihab, a Shari`ah researcher at Al-Azhar University, states,

As for performing tawaf on behalf of a sick or unable pilgrim,  it should be clear that a person cannot perform tawaf on behalf of a sick or unable person, because the sick person can perform tawaf in any convenient way; the sick or unable person may be carried, or may ride, to perform tawaf himself or herself.   Islam is based on easiness, so the sick or unable person is to look for a convenient way to perform his or her  tawaf.

The evidence for this is the hadith that shows that when Umm Salamah (may Allah be pleased with her) came to Makkah, she fell ill. On telling the Prophet (peace and blessings be upon him) about her illness, he told her to perform tawaf while riding. Also, Al-Bukhari and Muslim reported that the Prophet (peace and blessings be upon him) performed tawaf  in his  Hajj Al-Wadaa` (Last Hajj) while riding a camel.

Finally, we have to differentiate between the permissibility of performing the whole Hajj on behalf of an unable person and the impermissibility of performing tawaf on behalf of a pilgrim who is already attending the Hajj.