Nisaa
No, not without a barrier, in exactly the same way a menstruating woman may not touch the Mushaf during hayd. This is not a separate ruling invented for postpartum bleeding. It 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 the majority of scholars, across all four madhabs, apply the same restriction on touching the Qur'an to both.
The evidence is the same evidence used for hayd, and it reaches nifas by the same reasoning: the ayah (interpretation of the meaning): "None shall touch it except the purified." [Al-Waqiah 56:79], and the letter the Prophet (Salla Allahu 'Alayhi Wa Sallam) sent to the people of Yemen through 'Amr ibn Hazm, which stated: "No one should touch the Qur'an except one who is pure." (Narrated by Malik in Al-Muwatta. The chain has a gap, but al-Shafi'i and later scholars such as Ibn 'Abd al-Barr accepted it as supported by other routes, and it is what the majority of scholars act on.) Neither text names nifas specifically; both reach it the same way they reach hayd, through the shared absence of taharah the two states describe.
A minority of early scholars, among them al-Hasan al-Basri, Muhammad ibn Sirin, and Ibn Hazm, held that touching the Mushaf without purity is permitted in any state, reading the ayah as describing the Preserved Tablet (al-Lawh al-Mahfuz), which only the angels touch, rather than the physical copy in a woman's hands. The four madhabs did not adopt this view for hayd, and the same is true here.
What still reaches her
Touching the Mushaf through something separate from it, its cover, the bag it is carried in, a sleeve, is agreed across the schools to be permitted, in nifas exactly as in hayd. She is touching the barrier, not the Mushaf itself.
The Maliki concession for a woman who is teaching or learning Qur'an is written for exactly this case. It names hayd or nifas together: a woman in either state may touch and recite from the Mushaf out of necessity, since going without it for the length of her bleeding, which for nifas can run to forty or sixty days depending on the school, would disrupt her teaching or her studies in a way a shorter state would not. This is specific to teaching and learning, not a general permission to handle the Mushaf freely.
A Qur'an translation with no Arabic text splits the four madhabs the same way it does for hayd. The Hanafi school treats the Qur'an as its wording and its meaning together, so a translation still counts as Qur'an and the restriction still applies. The Maliki, Shafi'i, and Hanbali schools hold that the Qur'an's inimitability is specifically in its Arabic wording, so a translation on its own, without the Arabic, may be touched without purification.
A phone, tablet, or Qur'an app follows the same reasoning contemporary scholars apply to hayd: what the evidence restricts is touching a Mushaf, and a screen showing changeable pixels is not one, since it is glass and plastic rather than the fixed Arabic text a Mushaf is defined by. Islamweb's fatwa council put it in exactly those terms (fatwa no. 465004, 17 Rabi al-Awwal 1444 / 12 October 2022): purification is not required to hold the device, open the app, or scroll through it, though the fatwa still asks for basic respect, such as not reading from a screen that is visibly unclean. This is now the position of most contemporary scholars and fatwa councils, whether the state is hayd, nifas, or Janabah.
Reciting is a separate question
Reciting from memory or listening to Qur'an involves no touching at all, and is judged on its own evidence rather than this one. See can you recite Qur'an during your period for that separate difference of opinion, since nifas carries the same recitation ruling hayd does, for the same reason it carries this one.
Where the Maliki concession does not travel
It is worth noting what this concession does not do, since it clarifies why nifas gets it and Janabah does not. The concession rests on necessity: a state that runs for days and cannot be shortened at will. Nifas qualifies for exactly that reason, the same one that lets it borrow hayd's ruling in the first place. Janabah does not qualify, because Ghusl lifts it immediately, so no madhab records the same exception there.
When this restriction lifts
The restriction lifts the moment nifas ends and ghusl is performed, the same ghusl that reopens prayer, fasting, and entering the mosque. If bleeding continues past the recognised maximum, forty days on the predominant view, up to sixty for the Maliki and Shafi'i schools, as set out in how long nifas lasts, it is reclassified as istihadah rather than nifas, and istihadah does not restrict touching the Mushaf at all, the same way it does not restrict prayer or fasting.