While there are some broad families, even in the early Middle Ages, the structure and affiliation of manuscripts is not always clear. There are thousands of variants, some significant, some not. Some copyists tried to compare the texts available and create a better one, but they essentially conflated traditions and created new lines of descent. And manuscripts, even of the complete Bible, might take the text of individual books from different exemplars. Jerome produced three versions of the psalms.
Variants occur as well in early printed editions.
The latter edition, known as the Clementina, became the standard Latin Bible for the Catholic Church until the new edition in the 1980s. I don't know how exclusively official it was, although it may have been for liturgical purposes. But there are hundreds of editions. Latinam Vulgatam versionem ad codicum fidem, iussu Pii pp. XI, cura et studio monachorum Sancti Benedicti Commissionis pontificiae a Pio pp.