Answer
Wa`alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
During period, women can read the Quran and make dhikr and dua, though they are forbidden from fasting, performing Prayer and touching the mushaf.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
You are certainly allowed to engage in dhikr and du`aa‘ while menstruating. As Imam An-Nawawi pointed out, there is a virtual consensus among scholars and jurists that both men and women in a state of sexual defilement (janabah) and women while menstruating or undergoing a postpartum period of bleeding can make dhikr either through spoken words or silently in their hearts. It is also permissible for them, while making supplications, to use the familiar du`aa’s that are taken from the Qur’an such as the following: Rabbanaa hab lanaa min azwaajinaa wa dhurriyaatinaa qurrata a`yunin waij`alanaa li al-mutaqeena imaman (Our Lord, grant us true joy in our spouses and children and make us role models for those who are mindful [of You]). Rabbanaa aatinaa fi ad-dunya hasanatan wa fi al-aakhirati hasanatan wa qinaaadhaaba an-naar
(Our Lord, grant us good in this world and good in the next world and save us from the torments of the Hellfire).
Rabbanaa taqabbal minnaa innaka anta as-samee`u al-`aleem wa tub `alaynaainnaka anta at-tawwaab ar-raheem
(Our Lord, accept this work of ours, for verily You are All-Hearing, All-Knowing; and grant us repentance, for You are the Relenting, the Merciful).
Almighty Allah knows best.
Excerpted, with slight modifications, from: www.islam.ca.