This is one of the most important topics for women হাজীগণ. First, the reassuring news: menstruation does NOT invalidate your হজ্জ or prevent you from completing it. নবী করীম (সাল্লাল্লাহু আলাইহি ওয়া সাল্লাম) explicitly told Aisha when she began menstruating during her হজ্জ: 'Do everything that the হাজী does, except do not perform তাওয়াফ around the House until you are pure.'
What you CAN do while menstruating: stand at আরাফাত (the most essential pillar of হজ্জ), stay at মুযদালিফা, stone at জামারাত, make the animal sacrifice, remain in the state of ইহরাম, make unlimited dua and dhikr, attend the আরাফাত sermon, and collect pebbles. Regarding সাঈ, some scholars (including the Hanbali school) permit সাঈ during menstruation since it does not take place inside the মসজিদ proper. The farewell তাওয়াফ (তাওয়াফ al-Wada) is completely waived for menstruating women — this is unanimous among all scholars.
The only challenge is তাওয়াফ al-Ifadah, which is a pillar of হজ্জ and requires ritual purity according to the majority. Options include: (1) Wait until your period ends, purify, and perform তাওয়াফ — this is the safest approach if your schedule allows. (2) The Hanafi school permits performing তাওয়াফ al-Ifadah during menstruation with protection if departure cannot be delayed, but requires a camel sacrifice as penalty. (3) Consult a scholar about your specific situation.
Prevention strategy: many Muslim women use hormonal medication to delay their period during the হজ্জ days. Consult your doctor 2-3 months before departure to find a suitable option and trial it beforehand. This is permissible in Islamic law when done for a valid purpose like facilitating worship.