Praise be to Allah.
Firstly: in the cases where she is permitted to recite Qur’aan, she is also permitted to do Sajdat al-Tilaawah, if she happens to recite or hear a verse where the sajdah is prescribed. The correct view is that it is permissible for her to recite from memory, not from the Mus-haf, so on this basis the sajdah is prescribed for her, because it is not salaah (prayer), rather it is an act of submission to Allaah and worship, like other kinds of dhikr.
Secondly: the correct view is that it is not a condition for sajdat al-shukr and sajdat al-tilaawah on the part of one who is reciting or listening, that the person should be in a state of tahaarah (ritual purity), because they do not come under the rulings of salaah.
And Allaah is the source of strength. May Allaah bless our Prophet Muhammad and his family and companions.