Can Women Touch, Read, and or Recite the Qur'an
while in a state of impurity?

This is taken from the book on Dr.Bilal Phillips "Islamic Ruling on Menstruation and Post Natal Bleeding" page 12:

Touching and reciting the Qur’an. Menstruating women are permitted to read and touch the Qur’an.

*Footnote to the above statement: Although many scholars have prohibited menstruating women and people in the state of sexual defilement (junub) from reciting the Qur’an or touching it, Imaam al-Bukhaaree took the position that there was no authentic evidence to support such position and that it was in fact permitted. In support of his view he mentioned the positions of some of the early scholars who also permitted reciting and touching the Qur'an in order to show that the prohibition of menstruating women from reciting the Qur'an was not unanimous among scholars of the past. (Fat-h-al-Baaree, vol. 1 p. 305)

Ibn Taymeeyah said, "The prohibition of women from reading the Qur'an has no basis in either the Qur'an or the Sunnah. The statement attributed to the Prophet "The menstruating woman or person sexually defiled (junub) should not read anything of the Qur'an" is NOT authentic according to the unanimous opinion of the hadeeth scholars. Undoubtedly women used to menstruate during the Prophet's era and if recitation of the Qur'an was prohibited as formal prayer was prohibited, the Prophet would have explained it to his followers and his wives, and it would have been transmitted to us. Consequently, since no prohibition has been transmitted from the Prophet, it is not permissible to declare it haraam (forbidden) while knowing that he did not prohibit it. And if he did not prohibit it in spite of the many cases of menstruation during his time, it is obvious that it is not haraam
(al-Majmoo' vol. 26 p. 191)

Ibraaheem an-Nakha'ee (scholar among the students of the companions) said: "There is no harm in menstruating women reciting a Qur'anic verse. (It was also the opinion of Imam Maalik) (Vol. 1 p. 182 chap.. 9 of Sahih Al Bukhari) Imaam Al-Bukhari mentioned this incidence in order to prove that since the Prophet sent verses of the Qur'an to the pagans who were in a state of defilement, a menstruating woman should be able to read the Qur'an.

Fatawah or Islamic Rulings about some Important Issues by Sheikh Nasir-ud-Din-Al Albani
Q. - Is it prohibited to recite the Qur'an without wudoo?
A. - It is preferable to recite it upon purification. The hadeeth of the Messenger's making tayammum to reply to "as-Salaamu' alaikum" It is not forbidden to recite it without wudoo. As for the woman in her state of nifas or in after-birth confinement, then nothing prevents her from reciting the Qur'an. The hadeeth that At-Tirmidhee reported from Ibn 'Umar (r) that the Messenger said: "The woman ill with her period and the junub man may not recite the Qur'an" that is a 'munkar' hadeeth, as the Imaam of Sunnah Ahmad Ibn Hanbal says it is not authentic. Rather we have the hadeeth of Aisha (r) that the Messenger said to her when she made Hajj and she had her period: "Do everything the pilgrim does except prayer and tawaaf." Does not the pilgrim recite the Qur'an?"

So, this hadeeth is a clear proof of the permissibility of a woman's reciting/reading the Qur'an whilst in her period. Therefore it is not permissible to prevent the woman in her periods from reciting the Qur'an claiming that she is not pure! If Allah ta'ala had prevented her from reciting as He prevented her from praying, then we would follow that prevention and that would be the end of the matter. But He revealed to His Messenger a distinction between prayer and its constituent parts: since some people say that reciting the Qur'an is one of the pillars of the Prayer, so why do we not make purification a pre condition for it as it is for the prayer? So we say: Not only is reciting the Qur'an a part of the prayer, but also, prayer is begun by our saying "Allahu Akbar". So, we should also say that it is a requirement for saying 'takbeer' that we have wudoo! And the same goes for 'tasbeeh', 'tahmeed' etc. We find that some of the companions preferred to have wudoo to touch the Qur'an so that shows what is preferable, but to make it obligatory and to say that it is forbidden for a woman in her state of nifas to touch the Qur'an then that has no proof at all.

For a more indepth reading regarding this subject please go here:
The Requirement of Tahara for Reciting / Touching The Quran: