Firstly:

Most of the scholars are agreed that doing tawaf and sa‘i walking is better and preferable to doing them riding.

Ibn Qudaamah said: There is no difference of opinion regarding the fact that doing tawaf on foot – that is, walking – is preferable, because the Companions of the Prophet (blessings and peace of Allah be upon him) did tawaf walking, and other than in the Farewell Pilgrimage, the Prophet (blessings and peace of Allah be upon him) did tawaf walking.

End quote from al-Mughni (5/250).

Secondly:

They are agreed that it is permissible for one who has an excuse to do tawaf and sa‘i riding, whether the excuse has to do with sickness, disability, hardship, old age or the like.

That is because of the hadith of Umm Salamah (may Allah be pleased with her), who said: I complained to the Messenger of Allah (blessings and peace of Allah be upon him) that I was sick, and he said: “Do tawaf behind the people, riding.” [Agreed upon]

Al-Haafiz said: This indicates that it is permissible to do tawaf riding, if that is done because of a valid excuse. He only told her to do tawaf behind the people because that would be more concealing for her, and also so that she would not interrupt their rows and so that they would not be disturbed by her mount. End quote from Fath al-Baari (3/481).

Ibn Taymiyah said: It is permissible to do tawaf riding or being carried, when there is an excuse for that, according to the religious texts and scholarly consensus.

End quote from Majmoo‘ al-Fataawa (26/188).

And he said: If a person is not able to do tawaf walking, so he does tawaf riding or whilst being carried, that is acceptable, according to scholarly consensus.

End quote from Majmoo‘ al-Fataawa (26/125).

Thirdly:

With regard to doing tawaf or sa‘i riding when there is no excuse for that, this is a matter concerning which there is a difference of opinion among the scholars.

The Shaafa‘is and Zaahiris are of the view that the tawaf and sa‘i of one who rides is valid in all cases, because what is required is to circumambulate the Ka‘bah and to go between as-Safa and al-Marwah, and no matter how it is done, whether walking or riding, that fulfils what is required.

Moreover, al-Bukhaari (1530) and Muslim (1272) narrated that Ibn ‘Abbaas (may Allah be pleased with him) said: The Prophet (blessings and peace of Allah be upon him) did tawaf during the Farewell Pilgrimage on a camel.

The Prophet (blessings and peace of Allah be upon him) only did that for a reason, which was so that the people would be able to see him and ask him questions.

In Saheeh Muslim (1273) it is narrated that Jaabir said: The Messenger of Allah (blessings and peace of Allah be upon him) circumambulated the House during the Farewell Pilgrimage, on his mount, touching the Stone with his crooked staff, so that the people could see him and so that he could see them, and so that they could ask him questions, for the people had crowded around him.

An-Nawawi said: This explains the reason why he (blessings and peace of Allah be upon him) was riding.

End quote from Sharh Saheeh Muslim (9/19).

With regard to the report narrated in Sunan Abi Dawood (1881) via Yazeed ibn Abi Ziyaad, from ‘Ikrimah, from Ibn ‘Abbaas, that the Messenger of Allah (blessings and peace of Allah be upon him) was sick when he came to Makkah, so he did tawaf on his mount, it is a da‘eef (weak) report that cannot be relied upon to explain that the reason why he rode was because he was ill.

Among those who classed it as da‘eef were: al-Bayhaqi in Ma‘rifat as-Sunan wa’l-Athaar (7/258); Ibn Katheer in al-Bidaayah wa’n-Nihaayah (5/142); ash-Shawkaani in Nayl al-Awtaar (5/122); al-Mubaarakfoori in Tuhfat al-Ahwadhi (3/298); and al-Albaani in Da‘eef Sunan Abi Dawood.

Imam ash-Shaafa‘i said: Jaabir narrated from the Prophet (blessings and peace of Allah be upon him) that he did tawaf riding, and he said that he only did that so that the people could see him. This indicates that he did not do tawaf [riding] because he was sick, and I do not know that he (blessings and peace of Allah be upon him) fell sick during that pilgrimage of his.

End quote from al-Umm (3/443)

Ibn ‘Abbaas said: The people had crowded around the Messenger of Allah (blessings and peace of Allah be upon him), saying, This is Muhammad, this is Muhammad, until even the adolescent girls came out of their houses. People were not beaten to make way for the Messenger of Allah (blessings and peace of Allah be upon him), so when they crowded around him too much, he rode, but walking and walking rapidly are better. Narrated by Muslim (1264).

This view was favoured by Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him). He was asked about someone who rode in tawaf or sa‘i without needing to do so; what should he do?

He said: His tawaf is valid and his sa‘i is valid, because the Prophet did tawaf [riding] when he was in good health. As for the hadith of Ibn ‘Abbaas which says that he [the Prophet (blessings and peace of Allah be upon him)] fell sick and that is why he rode, it is da‘eef (weak), because its isnad includes Yazeed ibn Abi Ziyaad. The correct view is that he rode in order to ward off hardship from the people, lest they be harmed or harm him [as they crowded around him]. End quote. [Side A of tape 4 of the shaykh’s commentary on Kitaab al-Manaasik [the chapter of pilgrimage rituals] from Akhbaar al-Mustafa salla Allahu ‘alayhi wa sallam.

The majority of scholars are of the view that it is not allowed to do tawaf and sa‘i riding when there is no excuse. Some of them stated that a fidyah is required if a person does that and is not able to repeat it; others regard the tawaf and sa‘i as invalid and say that they must be repeated, as is the view of the Hanbalis.

Ibn al-Jawzi said: They differed concerning one who does tawaf riding with no excuse. There are two reports from Ahmad, one of which is that it is acceptable and he does not have to offer a compensatory sacrifice; this is also the view of ash-Shaafa‘i.

The other view is that it is not acceptable.

Abu Haneefah and Maalik said: It is acceptable, but he must offer a sacrifice. End quote from Kashf al-Mushkil min Hadith as-Saheehayn (2/433).

Ibn Qudaamah said: As for doing tawaf whilst riding or being carried, with no excuse, what may be understood from the words of al-Khiraqi is that it is not acceptable. This is one of the views narrated from Ahmad… because it is an act of worship that is connected to the Ka‘bah, so it is not acceptable to do it whilst riding with no excuse, like prayer. The second view is that it is acceptable, but a compensatory sacrifice must be offered. This is also the view of Maalik and Abu Haneefah. Al-Mughni by Ibn Qudaamah (5/250).

The ruling on sa‘i and tawaf is the same, according to the majority of scholars. Al-Mardaawi said: Doing sa‘i whilst riding is like doing tawaf whilst riding, according to the more correct view of the madhhab. End quote from al-Insaaf (4/13).

Conclusion:

The person who does tawaf and sa‘i should do them walking, unless he has an excuse. That is preferable according to scholarly consensus. If he does tawaf or sa‘i whilst riding with no excuse, then we have explained above that there is a difference of scholarly opinion concerning that.
What appears to be the case is that his tawaf and sa‘i are valid, in sha Allah.
However, it is not appropriate for a person to do that in the first place, so as to be on the safe side with regard to his worship, and so as to avoid a matter concerning which the scholars differed.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Some people are careless regarding that, and they do sa‘i in wheelchairs with no excuse, even though many of the scholars have said that doing sa‘i whilst riding is not valid without any excuse.

This matter is a matter concerning which the scholars differed, namely: with regard to sa‘i, is it stipulated that the individual must do it walking unless he has an excuse, or is it not?
But the individual should be prudent with regard to religious matters, and he should do sa‘i walking so long as he is able to do so, and if he is not able to do that, then Allah does not burden any soul with more than it can bear. End quote from Majmoo‘ al-Fataawa (22/451).

And Allah knows
best.