There is no report from the Prophet (peace and blessings be upon him) concerning du`as and adhkar to be recited in tawaf. You may offer whatever supplication you like or recite any dhikr.

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.

These are some of the conditions of tawaf: being Muslim, being of sound mind, forming the intention, covering the `awrah, being in a state of purity, performing seven complete circuits, keeping the Ka`bah on one’s left, performing tawaf on foot, if able to walk, and the circuits must be done consecutively.