The eminent Muslim scholar and renowned Da`iyah, Sheikh `Abdel Khaliq Hasan Ash-Shareef, states: “A menstruating woman is allowed to recite the Qur’an online as you said. This is in tune with the opinion of many Muslim scholars including Imam Ibn Al-Qayyim in his I`lam Al-Muqqi`in `an Rabb Al-`alamin.”
Elaborating more on this issue, we would like to cite for you the fatwa issued by the Saudi House of Fatwa headed by the late Sheikh `Abdul-`Azeez Ibn Baz, may Allah bless his soul:
“There is nothing wrong with menstruating woman or a woman in post-natal bleeding to recite the Qur’an, because there is no clear-cut authentic hadith that forbids them from doing so. However, it is reported in an authentic hadith that one who is junub (one in a state of impurity following sexual intercourse or wet dream), must not read the Qur’an while he or she is impure, according to the hadith reported by `Ali (may Allah be pleased with him).
The hadith reported as regards the menstruating woman and the one experiencing post-natal bleeding reads: ‘The menstruating woman and the one who is junub are not to read Qur’an.’ This hadith is reported by Ibn `Umar, but it is da`eef (weak), because the hadith was reported by Isma`eel Ibn `Ayyash from the Hijaziyeen, and he is famous for narrating da`eef hadiths from them.
However, such woman (in menstruation or post-natal bleeding) should not touch the mushaf (a copy of the Qur’an); she can only read from her own memory. As for the junub, he or she is not to recite the Qur’an, whether from memory or from the mushaf, until he or she has performed ghusl (purificatory bath). The difference between them is that the time span for the one who is junub to have himself or herself purified is very short; he or she can do ghusl right away after lovemaking or wet dream. The junub does not stay in this condition for long, and it is up to him/her when he/she wants to make ghusl; if he/she does not find water, he/she can do tayammum (dry ablution) and then he or she can pray and read the Qur’an. But the woman in menses or in post-natal bleeding has no control over her situation – the matter is up to Allah the Almighty.
Therefore, it is permissible for them to recite the Qur’an so that they do not forget it and they do not miss learning the teachings of Shari`ah from the Book of Allah. If that is the case, then it should certainly be permissible for them to read books containing du`a’ (supplication) that are mixed with verses and hadiths, etc. This is the view believed to the most correct.”
According to the above, it is permissible for a women during her menses to recite the Qur’an from the internet.