The oldest complete (that is, ‘containing many of the same books that we have now’) Biblical manuscripts all come from the 4th and 5th centuries: Codex Vaticanus (Vatican Library), Codex Sinaiticus (Russian National Library, British Library, St. Catherine’s Monastery at Mt. Sinai, and Leipzig University), and Codex Alexandrinus (British Library).
I know that the Vulgate was early - Do the originals still exist?
If by ‘original’ you mean ‘St. Jerome’s autograph’, then no. The earliest surviving copy of the Vulgate we have is Codex Amiatinus, made in the 8th century, about three centuries after Jerome. Codex Fuldensis (around 545) contains most of the New Testament, but the Gospels appear in a harmonized form.
How about an original Guttenberg?
As of now, there are 48 copies known to exist, 21 of which are complete. And that’s only the 42-line edition!