The position of the four madhhabs and the vast majority of scholars is that one must have wuḍūʾ to touch the muṣḥaf itself, based on Allah's words "None touch it except the purified" (Quran 56:79) and the Prophet's ﷺ letter to ʿAmr ibn Ḥazm, "No one is to touch the Qur'an except one who is pure" (Muwatta Malik, Book 15, Hadith 1, and recorded by others; its chain is mursal but is strengthened by supporting narrations, which is why the four schools acted on it). A minority, including Ibn Ḥazm, held that wuḍūʾ is not required for touching, so the matter has legitimate ikhtilāf — but the safest and majority view is to make wuḍūʾ. Either way, you may recite from memory, listen, read a translation, or read the Qur'an from a phone or tablet screen without wuḍūʾ, since the screen is not the muṣḥaf. One in a state of major impurity (janābah) should not recite the Qur'an until they perform ghusl, according to the majority.
Q&A · New Muslim
Can I touch or carry a copy of the Quran if I don't have wudu?
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.