There is legitimate ikhtilāf here. The majority of the four madhhabs hold that a menstruating woman should not recite the Qur'an, but the narration relied upon for the prohibition is graded weak by many hadith scholars; consequently a number of scholars (such as al-Bukhārī, Ibn Taymiyyah and al-Albānī) permit her to recite from memory, since there is no authentic text forbidding it and ʿĀ'ishah reported that the Prophet (ﷺ) would remember Allah at all times (Sahih Muslim 373). As for physically touching the muṣḥaf, most scholars require ritual purity on the basis of "none touch it except the purified" (Qur'an 56:79), so the safer practice is to hold it with a barrier such as a cloth or to read from a screen. In short: reciting from memory is permitted according to a strong view, while touching the muṣḥaf directly is best avoided out of caution.
Q&A · Women in Islam
Can a menstruating woman recite the Qur'an or touch the muṣḥaf?
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.