All praise be to Allah; and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Ritual purity isn`t a condition for performing the Hajj rites, save for the act of Tawaf (Hajj/Umrah). This is because being clean from major and minor impurities is a condition for the validity of the Tawaf. However, it is a condition for staying in Al-Masjid Al-Haram and other mosques to be in a state of ritual purity from a major impurity. In this regard, Fatwa differs according to time, place, and the legal considerations of the Muslim scholars.
If the menstruating woman could stay in Makah away from her journey companions, or was able to return to her destination and stayed in a state of Ihram till returning to Makkah again to complete the remaining rites when she becomes pure, then it is obligatory that she does so to eschew the concessions granted by the Muslim scholars, which we tend to resort to at times of extreme difficulty. However, if she had enough money, felt secure, had a Mahram(a husband or an unmarriageable relative) and the like, then, in principle, she should adhere to what the Prophet(PBUH) said to Aisha (may Allah be pleased with her) when she had her monthly period: " Perform Hajj rites as anyone who performs Hajj does them but do not perform Tawaf."[Reported by Imam Bukhari and Imam Muslim].
However, if she experienced extreme difficulty and couldn`t fulfill the above obligation, then the Muslim scholars have differed about the concession to be granted to her, each according to his Madhab(school of thought/juristic school):
First: The Shafi School of Thought If she fears to delay her journey, or fears for herself, then she could leave. If she reached a place from where she couldn`t return to Makka-some scholars said the distance (more than 83 Km.) for which a traveler is permitted to shorten his prayer- then she has to get out of her state of Ihram-as the one who was prevented from completing the Hajj rites (Ihsar)- by sacrificing a sheep and shortening her hair. However, she remains liable for Tawaf once she returns to Makkah-even after a long period of time- for this is the accredited view of the Shafi Madhab. However, they added that she should follow the Madhab of Abu Hanifa in order to be on the safe side.
Second: The Hanafi School of Thought The Hanafi scholars said that if she wore a nappy and performed the Tawaf, then her Tawaf is unlawful and involves a sin. Although this Tawaf gets her out of the state of Ihram, she must sacrifice a she-camel as an expiation.
Third: The Maliki School of Thought The Maliki scholars said that she may perform Tawaf during the purity interval, within her menses, because it is considered a state of purity as well. This is the accredited view of the Maliki Madhab as well as some Shafi scholars, although it isn`t the accredited view of the latter`s Madhab.
On his part, Ibn Taymia permitted the menstruating woman, who fears missing her journey, to perform Tawaf in while in that state of impurity and rendered her free from any liability. And Allah knows best.