Hic oracio non est in Hebraica ueritate, nec in Regum nec in Paralipomenon, sed hic interserit eam Ieronimus (“this prayer is not in the Hebrew truth, neither in Kings nor Chronicles, but Jerome inserted it here”).Let me provide some context.
The Prayer of Manasseh is one of those documents in the Vulgate Appendix. It was excluded from the biblical canon by the Council of Trent in 1546, even though it had occupied a place in biblical manuscripts and editions of the Vulgate prior to the Council.
(Actually, I'm not exactly sure how we know that the Council excluded the Prayer. I mean, the Council doesn't mention it in the canon list, but then I don't guess you'd really expect the Prayer to get an independent mention if it's thought of as only an appendix to Chronicles. After all, Lamentations wasn't mentioned by the Council either, but I don't think anyone suspects that the Council intended to exclude Lamentations. It was just considered an appendix to Jeremiah. At any rate, the Prayer has not been considered a part of the Roman Catholic biblical canon established by the Council, while Lamentations has been so considered.)
You can see that Gutenberg located the Prayer immediately after Chronicles (fol. 226r), though here it doesn't appear so much as an appendix as a separate work, similar to the way Esdras appears just afterwards. Same in this edition of the Vulgate edited by Jan Henten with a date of 1583 (first published 1547, described by Gordon and Cameron in NCHB 3, pp. 192–93). And in the Vatable Bible published by Stephanus 1545 (the link is to vol. 2; see vol. 1 here; described by Gordon and Cameron on p. 191).
On the other hand, here's an example of a Paris Bible (Beinecke Yale MS 793) from the thirteenth century, and the Prayer does appear without any break as the conclusion to Chronicles. The Prayer starts on fol. 210r, at the bottom right. You can see that someone later has marked the start of the Prayer, but originally its text was continuous with Chronicles. An explicit appears on the other side of the page after the Prayer, and an inicipit introduces Esdras (or, actually, Jerome's Prologue to Esdras, with the biblical text beginning on the next page).
In this next example, Paris BNF latin 15467 from the year 1270, it is even harder to distinguish between the end of Chronicles and the Prayer. The Prayer begins in the middle of line 13 in the left column of image 220. Again, immediately after the Prayer, Jerome's Prologue to Esdras starts at the top of the next column.
Anyway, from what I hear, the Prayer of Manasseh started to appear in Latin Bibles only in about the 13th century, so these Paris Bibles are early examples, at least as preserved.
What about before the 13th century?
Well, the Prayer was translated into Latin from Greek, of course. Now, if you pick up your Rahlfs(-Hanhart) edition of the LXX, you will not find the Prayer listed in the Table of Contents, but you will find a work called the Odes immediately after the Psalms. The Odes consists mostly of excerpts from other parts of the Bible: Ode 1 is the Song of the Sea (Exod 15), Ode 2 is the Song of Moses (Deut 32), Ode 6 is the Prayer of Jonah (Jonah 2), Odes 7–8 are the (deuterocanonical) Prayer of Azariah and the Song of the Three Young Men (Daniel 3). And Ode 12 is the Prayer of Manasseh.
(The essential book now on the Odes is by Marguerite Harl (2014). The sequence of the odes is arbitrary, and differs according to manuscripts. The earliest surviving manuscript to include the Odes is Alexandrinus. You can take a look here = CSNTM's digital images of the nineteenth-century facsimile by Thompson. The Prayer is Ode 8 in Alexandrinus, and starts at the bottom right of what is labeled in the manuscript as fol. 567 = image GA_02_0557a.jpg.)
It was as part of a similar collection of odes (cantica) that the Prayer of Manasseh first came to be known in Latin. While we know that such collections of canticles were around in Latin from at least the fourth century, the Prayer of Manasseh is clearly attested only from the sixth century, in the commentary on the Canticles by Bishop Verecundus of Junca, such an important author that he merits two separate Wikipedia pages.
But Stephen Langton thought that Jerome was responsible for locating the Prayer after Chronicles, as you can see from the quotation at the top of this post. Langton (main subject of only a single Wikipedia page) is most famous generally for his role in the situation leading to the Magna Carta, but he is also well-known to biblical scholars for popularizing our present chapter divisions. He wrote many biblical commentaries, of which few have been printed. But the beginning of his commentary on the Prayer of Manasseh has been printed by Beryl Smalley in G. Lacombe and B. Smalley, “Studies on the Commentaries of Cardinal Stephen Langton,” Archives d’histoire doctrinale et littéraire du moyen âge 5 (1930): 1–220, on p. 158 (available here). The comment I quoted at the beginning of this post can be found there.
I'm left with some questions? Did Langton not know about Bibles that excluded the Prayer? Did he not realize that it was only during his own lifetime that the Prayer came to be located after Chronicles? Or did he think that this recent habit restored a long-abandoned practice introduced by Jerome?
Just for the record, we have no evidence suggesting that Jerome was familiar with the Prayer of Manasseh.