There are at least 5, however incomplete, 1st/7th-century manuscripts, and they include the following:
1) DAM 01-27.1 (last half of the 1st/7th century CE—that is, between 650 and 685 CE)
2) Arabe 328a-b and Marcel 18 (third quarter of the 1st/7th century CE—that is, between 671 and 695 CE)
3) Arabe 328c and Mingana Islamic Arabic 1572a (end of the 1st/7th century and beginning of the 2nd/8th century CE)
4) DAM 20-33.1 (during the reign of al-Walīd b. ʿAbd al-Malik [r. 705–715])
5) Or. 2165 (end of the 1st/7th century and beginning of the 2nd/8th century CE)
6) Codex Amrensis 1 (first half of the 2nd/8th century)
7) Samarkand Codex (beginning of the 2nd/8th century CE)
8) Cairo Codex (end of the first quarter of the 2nd/8th century CE)
9) H.S. 44/32 (second quarter of the 2nd/8th century CE—that is, between 661 and 750 CE)
M. Lamsiah, Makhṭūṭāt al-Qurʾān: madkhal li-dirāsat al-makhṭūṭāt al-qadīma
, Canada, 2017, p. 82.
One should also mention the “Gotthelf Bergsträsser Archive”, which consisted of 154 microfilms of very old Koran codices. Unbeknownst to most, these materials did not perish in the American bombardment during World War II but were in fact saved and stored in a secure location by a German Arabist, Anton Spitaler (1910–2003). Since 1990, these materials have been in the possession of German Angelika Neuwirth, co-founder of the Corpus Coranicum project. Only recently has Neuwirth published these microfilms. And if some of these materials are indeed ancient, one should also add them to our list of oldest Korans.
In 2001, the late S. Noja Noseda, professor of Arabic Language and Literature at the Università Cattolica in Milan, alongside François Déroche, a specialist of Arab manuscripts at the National Library of France, analyzed the contents of all hijazid manuscripts securely dated to the seventh century. By comparing the manuscripts to the King Fuʾād edition, they concluded that 83 % of the Koran is represented in those manuscripts. Note, however, that Noseda & Déroche did not include in their analysis Koran materials written on papyri, inscriptions, nor the famous Sanaa palimpsest (Sanaa, Inv. 01-27.1). Taking this into account, and considering a more recent analysis by islamic-awareness.org, the number given by Noseda & Déroche must be higher, somewhere around 90 %.
In conclusion, despite not having found (in the seventh century, yet) a complete codex, we can say with certainty based on our hijazid manuscripts, inscriptions, papyri, that at least 90 % (or 83 %, going Noseda & Déroche's outdated estimate) by of the Koran existed by the middle of the seventh century (the 650s).
For my brief text on the Quran, see:
F. Déroche and S. N. Noseda (Eds.), Sources de la transmission manuscrite du texte coranique. I. Les manuscrits de style hijazi. Volume 2. Tome I. Le manuscrit Or. 2165 (f. 1 à 61) de la British Library, 2001, Fondazione Ferni Noja Noseda, Leda, and British Library: London, p. xxvii.
Nicolai Sinai, The Qur'an: A Historical-Critical Introduction (Edinburgh University Press, 2017), pp. 45-46.
Patricia Crone, "Foreword", in The Qurʾānic Pagans and Related Matters, ed. Hanna Siurua (Brill, 2016), p. xiii.https://www.islamic-awareness.org/quran/text/mss/hijazi.html