The majority of scholars agree that ritual purity (taharah) is not a condition for the validity of السعي between الصفا and المروة. This is because السعي is not performed inside the المسجد الحرام itself, and النبي (صلى الله عليه وسلم) told Aisha to do everything that a الحاج does except الطواف around the الكعبة. Since السعي was not excluded, it indicates that purity is not required for it.
الشيخ ابن باز confirmed that السعي is valid without الوضوء and without الغسل, stating that purity is مستحب (مستحب) but not required for السعي. However, he clarified an important practical point: since السعي must come after a valid الطواف, and الطواف requires purity, a menstruating woman should first wait until she is pure, perform الغسل, do her الطواف, and then perform السعي. She cannot perform السعي before doing her الطواف.
الشيخ ابن عثيمين added that in the case of الحج, if a woman has already performed الطواف al-Qudum before her menses began and has not yet done السعي, she may perform السعي even while menstruating, since it follows a valid الطواف. The key distinction is that السعي itself does not require purity, but its prerequisite — الطواف — does.