Nisaa
Yes. Istihadah does not stop her from reciting Qur'an, and this one does not even wait on her wudu the way touching the Mushaf during istihadah does.
The restriction some scholars place on recitation is built specifically on hadath akbar, major ritual impurity. The Hanafi, Shafi'i and Hanbali schools forbid a menstruating woman from reciting by comparing hayd to Janabah, citing the hadith of 'Ali (Radi Allahu Anhu) that the Prophet (Salla Allahu 'Alayhi Wa Sallam) used to recite Qur'an to them in every state, and nothing stopped him from it except Janabah (reported by Abu Dawud, Tirmidhi, al-Nasai, Ibn Majah, Ahmad and Ibn Khuzaymah; graded hasan sahih by al-Tirmidhi). See can you recite Qur'an during hayd and can you recite Qur'an during Janabah for how that same hadith plays out differently in each of those two states, and where Imam Malik and Ibn Taymiyyah part ways with the majority specifically on hayd. Nifas follows hayd's position on this exactly; see can you recite Qur'an during nifas for why.
Istihadah never enters this debate in the first place, because it is not hadath akbar to begin with. As already established for her prayer, her fasting, her tawaf, the mosque and touching the Mushaf, a woman with istihadah is treated as someone with a continuous minor hadath, comparable to chronic incontinence, on the strength of what the Prophet (Salla Allahu 'Alayhi Wa Sallam) told Fatimah bint Abi Hubaysh (Radi'Allahu Anha) when she asked whether she should abandon prayer: "No, that is from a vein and not menstruation." (Narrated by al-Bukhari and Muslim). None of the four madhabs analogise a continuous minor hadath to Janabah the way they analogise hayd to it, so the comparison that grounds the restriction on recitation during hayd and Janabah has nothing to attach to here.
Why this is lighter even than touching the Mushaf
Reciting Qur'an aloud or from memory does not require wudu at all, for anyone, as the ordinary rule. Wudu is what the majority require specifically for touching the physical Mushaf, not for recitation itself; see can you touch the Qur'an during istihadah for that separate requirement and the pad-and-wudu routine that opens it to her within each prayer's time. A person carrying ordinary minor hadath, someone who simply has not yet made wudu since waking, may still recite Qur'an freely; only touching the Mushaf and standing for prayer wait on wudu. Istihadah does not change that baseline. She may recite at any time, including in the gap between prayers before she has renewed her wudu for the next one.
That leaves the Mushaf itself as the only part of this still tied to her wudu. She may recite from memory or aloud without it, but still needs the same routine set out in can you pray during istihadah before touching the physical pages, or she can hold it through a barrier, or read from a phone or Qur'an app, in exactly the way that same question on touching the Mushaf sets out for hayd, nifas and Janabah alike.
If the bleeding might actually be hayd rather than istihadah, telling the two apart first is what decides whether this answer, or the more restrictive one that applies during hayd, is the one that reaches her.