Do You Need Wudu for Tawaf and Sa`i?
Question

During tawaf when doing ‘Umrah I broke my wudu, and I did not know what to do, so I went out and did wudu, then I repeated tawaf and did aa’i between al-Safa and al-Marwah. Is what I did correct? What should I have done?
Praise be to Allah.Is tawaf without ablution valid?
You did well by renewing your wudu and repeating your tawaf. You did that which is better and more on the safe side. The majority of scholars are of the view that taharah (being pure, i.e., having wudu) is an essential condition of tawaf, like prayer, being valid. Just as prayer is not valid unless one has done wudu, so too tawaf is not valid without wudu. 
Ibn Qudamah said: 
“Taharah (being pure, having wudu) is a condition of tawaf being valid. This is the well-known view of Ahmad, and it is the view of Malik and al-Shafi'i.”
Evidence for wudu being a condition for tawaf
The majority of scholars quote several pieces of evidence to support this view, such as the following: 

The Prophet (peace and blessings of Allah be upon him) said: “Tawaf around the House is (a kind of) prayer, except that you may speak during it.” (Narrated by al-Tirmidhi, 960; classed as sahih by al-Albani in Irwa al-Ghalil, 121) 
It is proven in al-Sahihayn that ‘Aishah (may Allah be pleased with her) said: “When he [the Prophet] (peace and blessings of Allah be upon him) wanted to do tawaf, he did wudu.” And he (peace and blessings of Allah be upon him) said: “Learn from me your rituals (of Hajj and ‘Umrah).” (Narrated by Muslim, 1297) (Fatawa al-Shaykh Ibn Baz, 17/213-214)
It was proven in al-Sahihayn that the Prophet (peace and blessings of Allah be upon him) said to ‘Aishah when her menses came: “Do what the pilgrims do, but do not circumambulate the House until you are pure [i.e., until your period ends].” 

Any expiation for making tawaf without wudu?
Shaykh Ibn Baz (may Allah have mercy on him) was asked: I have a female relative who did ‘Umrah during Ramadan and when she entered the Haram she broke her wudu by passing wind, but she felt too embarrassed to tell her family that she wanted to do wudu. Then she did tawaf and when she finished tawaf she went off by herself and did wudu, then she did sa’i. Does she have to offer a sacrifice or any expiation (kaffarah)? 
He replied: 
“Her tawaf is not valid, because one of the conditions of tawaf being valid is taharah (purity), as in the case of prayer. So she should go back to Makkah and circumambulate the House. It is also recommended for her to repeat sa’i, because most of the scholars are of the view that it is not permissible to do sa’i before tawaf. Then she should cut some hair from all parts of her head and exit ihram. If she has a husband who had intercourse with her, then she has to slaughter a sheep in Makkah to be distributed to the poor, and she has to do another ‘Umrah from the miqat from which she entered ihram for the first ‘umrah, because the first ‘umrah was invalidated by that intercourse. She should do what we have mentioned, then do another ‘Umrah from the mهqat from which she entered ihram for the first ‘umrah, whether she does that now or at another time, depending on her ability. And Allah is the source of strength.” (Fatawa al-Shaykh Ibn Baz, 17/214-215)
He was also asked: 
A man started to do tawaf, then he passed wind. Does he have to stop his tawaf or should he continue? 
He replied: 
“If a person passes wind or urine or semen, or he touches his private part, etc. during tawaf, then he should stop his tawaf, as (he is required to do if any of these things happen) in the case of prayer. He should go and purify himself, then start tawaf again. This is the correct view; there is a difference of scholarly opinion concerning this matter, but this is the correct view concerning both prayer and tawaf, because the Prophet (peace and blessings of Allah be upon him) said: “If one of you breaks wind silently during prayer, let him go and do wudu then repeat the prayer.” Narrated by Abu Dawud and classed as sahih by Ibn Khuzaymah. Tawaf comes under the same heading as prayer in general…” (Majmu’ Fatawa al-Shaykh Ibn Baz, 17/216-217)
Some of the scholars are of the view that purity from hadath (minor impurity) is not essential for tawaf. This is the view of Abu Hanifah (may Allah have mercy on him) and was the view favoured by Shaykh al-Islam Ibn Taymiyah. They responded to the evidence quoted in support of the first view in the following manner: 
With regard to the hadith: “Tawaf around the House is (a kind of) prayer”, they said that this is not a sahih report from the Prophet (peace and blessings of Allah be upon him), rather these are the words of Ibn 'Abbas (may Allah be pleased with him). Al-Nawawi said in al-Majmu’: The correct view is that this is a mawquf report which stops at Ibn 'Abbas. This was mentioned by al-Bayhaqi and other scholars of hadith. 
With regard to the actions of the Prophet (peace and blessings of Allah be upon him) and his doing tawaf in a state of taharah (purity), they said: This does not indicate that it is obligatory, rather it only indicates that it is recommended, because the Prophet (peace and blessings of Allah be upon him) did that, but it is not reported that he commanded his Companions to do that. 
With regard to the Prophet’s saying to ‘Aishah: “Do what the pilgrims do, but do not circumambulate the House until you are pure [i.e., until your period ends],” the Prophet (peace and blessings of Allah be upon him) told her not to do tawaf because she was menstruating, and the menstruating woman is forbidden to enter the mosque. 
Shaykh al-Islam Ibn Taymiyah said: 
“Those who say that it is obligatory to have wudu in order to do tawaf have no proof at all, because no one narrated, with either a sahih isnad or a da’if one, that the Prophet (peace and blessings of Allah be upon him) enjoined having wudu in order to do tawaf. It should be noted that very many people did Hajj with him, and he did ‘Umrah numerous times and the people did ‘Umrah with him. If wudu were obligatory for tawaf, the Prophet (peace and blessings of Allah be upon him) would have pointed that out to everyone, and if he had stated that, then the Muslims would have transmitted it from him and would not have been negligent concerning that.  But it is proven in al-Sahih that when he did tawaf he did wudu. This on its own does not indicate that it is obligatory, for he used to do wudu for each prayer and he said: “I do not like to remember Allah except in a state of purity.”  (Majmu’ al-Fatawa, 21/273)
Even though this view –that taharah is not an essential condition for tawaf – has some strength and the evidence may be interpreted in this manner, people should not start tawaf without being in a state of purity, because it is undoubtedly better to do tawaf in a state of purity; that is more on the safe side and by doing that one will avoid going against the view of the majority of scholars. 
But a person may do that if it is too difficult to keep or repeat one's wudu, such as during the days of Hajj, or if a person is sick or elderly and it is hard to keep his wudu as well as deal with the crowds and so on. 
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said, after responding to the evidence quoted by the majority: 
“Based on this, the more correct view which gives peace of mind is that it is not essential to be pure from minor impurity (i.e., have wudu) in order to do tawaf, but it is undoubtedly better and is following the example of the Prophet (peace and blessings of Allah be upon him), and no one should neglect purity and go against the majority of scholars in this matter. But sometimes one is compelled to follow the view of Shaykh al-Islam Ibn Taymiyah, such as if a person breaks his wudu when there is too much crowding, in which case the view that he has to go and do wudu and come back in this severe crowding, especially when he only has a few circuits left, will cause too much hardship. With regard to matters which cause too much hardship and regarding which there is no clear text, we should not force people to adhere to them, rather we should follow that which is more lenient, because forcing people do things that are too hard with no clear proof contradicts what Allah says (interpretation of the meaning): 
“Allah intends for you ease, and He does not want to make things difficult for you.” [al-Baqarah 2:185] (Al-Sharh al-Mumti’, 7/300)
Is wudu necessary for sa`i?
With regard to sa’i, it is not essential to have wudu for sa’i . This is the view of the four Imams, Abu Hanifah, Malik, al-Shafi'i and Ahmad. Indeed it is permissible for menstruating women to perform sa’i between al-Safa and al-Marwah, because the Prophet (peace and blessings of Allah be upon him) only told menstruating women not to do tawaf, as he said to ‘Aishah (may Allah be pleased with her), when her menses came: “Do what the pilgrims do, but do not circumambulate the House.”  (See al-Mughni, 5/246)
Shaykh Ibn ‘Uthaymin said: 
“If a person does sa’i when he does not have wudu, or when he is junub [in a state of impurity following sexual activity], or if a woman does sa’i when she is menstruating, then this is valid, but it is better to do sa’i in a state of taharah.”  (Al-Sharh al-Mumti’, 7/310, 311)
For more about tawaf and sa`i, please see these answers: 109246 , 234172 , 106598 , 34644 , 36869 , 109274 and 253569
And Allah knows best.