This is one of the most important topics for women jamaah haji. First, the reassuring news: menstruation does NOT invalidate your Haji or prevent you from completing it. Nabi (shallallahu alaihi wa sallam) explicitly told Aisha when she began menstruating during her ibadah haji: 'Do everything that the jamaah haji does, except do not perform Tawaf around the House until you are pure.'
What you CAN do while menstruating: stand at Arafah (the most essential pillar of Haji), stay at Muzdalifah, stone at Jamarat, make the animal sacrifice, remain in the state of Ihram, make unlimited doa and dhikr, attend the Arafah sermon, and collect pebbles. Regarding sa'i, some scholars (including the Hanbali school) permit sa'i during menstruation since it does not take place inside the masjid proper. The farewell Tawaf (Tawaf al-Wada) is completely waived for menstruating women — this is unanimous among all scholars.
The only challenge is Tawaf al-Ifadah, which is a pillar of Haji and requires ritual purity according to the majority. Options include: (1) Wait until your period ends, purify, and perform Tawaf — this is the safest approach if your schedule allows. (2) The Hanafi school permits performing Tawaf 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 Haji days. Consult your doctor 2-3 months before departure to find a suitable option and trial it beforehand. This is boleh in Islamic law when done for a valid purpose like facilitating worship.