The majority of scholars agree that ritual purity (taharah) is not a condition for the validity of sa'i between Safa and Marwa. This is because sa'i is not performed inside the Masjidil Haram itself, and Nabi (shallallahu alaihi wa sallam) told Aisha to do everything that a jamaah haji does except Tawaf around the Kabah. Since sa'i was not excluded, it indicates that purity is not required for it.
Sheikh Ibn Baz confirmed that sa'i is valid without wudhu and without mandi wajib, stating that purity is sunnah (mustahab) but not required for sa'i. However, he clarified an important practical point: since sa'i must come after a valid Tawaf, and Tawaf requires purity, a menstruating woman should first wait until she is pure, perform mandi wajib, do her Tawaf, and then perform sa'i. She cannot perform sa'i before doing her Tawaf.
Sheikh Ibn Uthaymeen added that in the case of Haji, if a woman has already performed Tawaf al-Qudum before her menses began and has not yet done sa'i, she may perform sa'i even while menstruating, since it follows a valid Tawaf. The key distinction is that sa'i itself does not require purity, but its prerequisite — Tawaf — does.