At the time that Codex Sinaiticus was made, around the middle of the 4th century, there was no agreement as to which books constituted the Bible, or the order in which they should be arranged. Indeed, before this date there had been no concept of the Bible as a single volume, containing all those texts familiar to modern readers.
I would suggest firstly that the lack of complete agreement (as exhibited in minor differences within fourth century canon lists and differences in the order of books in extant manuscripts) should not be presented as if this implies that "there was no agreement as to which books constituted the Bible", since for the most part and for the core books there clearly was widespread agreement.
Secondly, even though it is the case that the order of books takes on a different significance when they are all bound in a single codex, conventions of ordering were clearly in place in the period before the fourth century. Such conventions of ordering can be seen in a variety of places both internal to the sub-collections which make up the NT (e.g. P46 for Paul; P45 for Gospels & Acts), and in discussions reflecting on the developing NT as a whole (Irenaeus, Muratorian Canon etc.). It is not random that the Gospels precede the apostolic witness. So yes there is no completely settled order for the NT books (nor is there after Sinaiticus since it is only printing that brings all the books together for the masses), but that does not mean there were no areas of general agreement in relation to the ordering of the books.
Thirdly, it is in principle very unlikely that Sinaiticus is the first manuscript of the Bible (both because Vaticanus is probably earlier, and because there may have been several hundred other earlier mss now lost). So we don't know whether a Bible in a single volume had actually been physically constructed well before Sinaiticus. If we are talking not about physical artefacts but about the capacity to conceptualise the various bits of the NT as a single collection then we might have several candidates.
So these points need a rather more nuanced presentation in my opinion.
On the manuscript itself we read the following:
To create Codex Sinaiticus required the collaboration of arguably four scribes (designated by modern scholars A, B1, B2 and D), working from approximately thirty exemplars. In order to expedite the process, the scribes would have copied different parts of the text at the same time, dividing the work between them. For example, Scribe D is known to have copied Genesis and portions of Matthew, Mark and Luke (in addition to other books of the Bible); while Scribe A produced large chunks of the Old Testament, and most of the New Testament, including Acts.
Firstly a minor problem here is the division of B into B1 and B2. So far there is no scholarly publication in any format outside the Sinaiticus project web-site that argues the case for the separation of B into two scribes. The case for this has never been made in a published forum (I don't recall it even being affirmed in Parker's book on Sinaiticus, although I could be wrong on that).
Secondly, this notion that the scribes worked from 'thirty exemplars' is rather interesting. I would love to see the evidence behind this assertion. I presume that doesn't mean 30 for the NT, as if every separate text had its own separate exemplar (that would not be plausible), but 30 for the whole Bible (including the bits of the OT that we no longer have). One could easily imagine 6-10 exemplars for the NT, but moving from imagining it to actually constructing an argument on the basis of some evidence would require some interesting work.
Thirdly, the notion that Scribe D's work on portions of Matthew, Mark and Luke is an example of dividing the work to expedite the process is clearly wrong; since scribe D is working to replace sheets within quires that are otherwise the work of scribe A. The author should have read Jongkind.