Nisaa
No, not in that state. Tawaf is the one part of Hajj and Umrah that hayd genuinely stops you from performing, in a way that fasting, prayer, or reciting du'a during the rest of the pilgrimage do not. Every other rite stays open to a menstruating woman; only the actual circling of the Ka'bah waits on purity, whether she is on Hajj, on Umrah alone, or simply visiting the Haram.
The clearest evidence is the Prophet's (Salla Allahu 'Alayhi Wa Sallam) own instruction to 'Aishah (Radi'Allahu Anha) during Hajjat al-Wada' (the Farewell Pilgrimage), after her period began while she was in ihram for Umrah: "Do everything that the pilgrims do, except do not perform Tawaf around the House until you are pure." (Narrated by al-Bukhari and Muslim). Everything else she was told to continue: standing at Arafah, spending the night at Muzdalifah and Mina, throwing the jamarat, du'a and talbiyah, one of the acts of worship that stay open during hayd. Tawaf alone was set aside. For the specific case of a period beginning mid-Hajj with a flight home already booked, see what if your period starts during Hajj, which covers that timing question on its own.
Why the majority tie tawaf to purity
The four madhabs all agree that a menstruating woman should not perform tawaf. Where they differ is on whether her purity is a shart, a condition without which the tawaf does not count at all, or a wajib, a strong obligation whose omission is sinful but does not by itself undo the tawaf.
The Maliki, Shafi'i, and Hanbali schools hold that purity is a shart. Their basis is a hadith of Ibn 'Abbas (Radi'Allahu Anhuma): "Tawaf around the House is like Salah, except that Allah has permitted speech during it; whoever speaks during it should say nothing but good." (Narrated by al-Tirmidhi; graded sahih by al-Albani, though al-Nawawi and others held that the correct view is that this is a statement of Ibn 'Abbas himself rather than words of the Prophet). Since tawaf is treated as equivalent to salah in everything except the permission to speak, and salah requires purity, so does tawaf on this reading. A tawaf performed while bleeding simply does not count, and she remains obligated to repeat it once she is pure, however long that takes.
Where the Hanafi school differs
The Hanafi school treats purity from hadath, hayd included, as wajib rather than a shart for tawaf's validity. On this view a tawaf performed while menstruating is valid but blameworthy, and a dam, traditionally a badanah (a camel or cow), becomes due as expiation for having performed it in that state. This is the position most contemporary fatwa councils reach for when a pilgrim genuinely has no way to extend her stay: it lets her leave Makkah with her tawaf actually completed rather than owed.
Tawaf al-Wada' is a separate case
The farewell tawaf done just before leaving Makkah is not owed by a menstruating woman at all, and no dam covers it, because it is not owed in the first place. Ibn 'Abbas (Radi'Allahu Anhuma) said that people were commanded to make the farewell tawaf their last act before leaving, except that this was waived for a menstruating woman. (Narrated by al-Bukhari and Muslim). That waiver is specific to Tawaf al-Wada'. It does not extend to the obligatory tawaf of Hajj or the tawaf of Umrah itself, the rite her period actually reaches, which the majority still requires her to make up once pure, and which the Hanafi school still treats as owed unless covered by a dam.
She also cannot enter the mosque to reach the Ka'bah while bleeding, for the same reason she would not enter any other mosque, which is part of what makes the wait for purity, rather than a workaround, the normal outcome for most women whose period simply runs its course before they need to fly home.
Because a delayed tawaf or an arranged dam both carry real, situation-specific consequences, this is worth confirming with a scholar directly, ideally before travel, if her period is likely to fall across her Hajj or Umrah days. For the specific timing question of a period beginning during an Umrah trip on its own, see what if your period starts during Umrah.