Nisaa
No, not while nifas continues, in exactly the same way a menstruating woman may not perform tawaf during hayd. This follows directly from the principle set out in are the rulings of nifas the same as hayd: the two states share the same essential nature, and tawaf is the one rite hayd's evidence stops outright, so nifas is held to the same limit.
The clearest sign that this is a genuine, separately recognised restriction, and not just a general caution around childbirth, is the hadith of Asma bint 'Umays (Radi'Allahu Anha). She began bleeding from childbirth during the Farewell Pilgrimage, and the Prophet (Salla Allahu 'Alayhi Wa Sallam) told her: "Bathe, bind yourself with a cloth, and enter ihram." (Narrated by Muslim). She was told to proceed with ihram and the rest of the rites while nifas continued; tawaf was the one part withheld, on the same basis 'Aishah (Radi'Allahu Anha) was told during her own period: "Do everything that the pilgrims do, except do not perform Tawaf around the House until you are pure." (Narrated by al-Bukhari and Muslim). Nifas reaches that instruction the same way hayd does, because both leave her outside the taharah tawaf requires.
Where the four madhabs differ
The disagreement is the same one that applies to tawaf during hayd, carried over unchanged. The Maliki, Shafi'i and Hanbali schools treat purity as a shart, a condition without which the tawaf does not count, resting on the hadith of Ibn 'Abbas (Radi'Allahu Anhuma): "Tawaf around the House is like Salah, except that Allah has permitted speech during it." (Narrated by al-Tirmidhi; graded sahih by al-Albani). A tawaf performed during nifas would need to be repeated once she is pure. The Hanafi school treats purity as wajib rather than a shart, so a tawaf performed during nifas is valid but blameworthy, and a dam, traditionally a badanah, a camel or cow, becomes due, the same concession contemporary fatwa councils reach for when a pilgrim genuinely cannot extend her stay; see what if your period starts during Hajj for how this plays out for hayd specifically.
Tawaf al-Wada' is waived, the same way it is for hayd
The farewell tawaf is not owed by a woman in nifas at all, for the same reason it is not owed by a menstruating woman. Ibn 'Abbas (Radi'Allahu Anhuma) said people were commanded to make the farewell tawaf their last act before leaving the House, "except that this was waived for a menstruating woman." (Narrated by al-Bukhari and Muslim). The waiver reaches nifas the same way the rest of hayd's rulings do. It does not extend to the obligatory tawaf of Hajj or the tawaf of Umrah itself, which the majority still requires her to make up once she is pure.
Why this matters for a new mother's travel plans
Nifas can run considerably longer than hayd, up to forty or sixty days depending on the school, as set out in how long nifas lasts. A woman planning Umrah soon after childbirth, or completing a Hajj or Umrah she began before delivering, should expect nifas to outlast a short trip, and plan around a delayed tawaf or the Hanafi dam rather than assume the bleeding will simply stop in time.
If nifas runs past the recognised maximum
Once bleeding passes her madhab's ceiling, it is reclassified as istihadah rather than nifas, and istihadah does not stop tawaf the way hayd or nifas does. She performs ghusl, resumes tawaf, and takes on the same pad-and-wudu routine any woman with istihadah follows before prayer, rather than waiting for the bleeding to stop entirely.
The same suspension covers her prayer, her fast, and the mosque for as long as nifas itself continues; see can a woman in nifas enter or stay in a mosque for what still reaches her while she waits.