What are the oldest copies we have of the Septuagint (LXX)? I know that it is present in Codex Sinaiticus and other codices. Are there any in scrolls and papyri or are there only codices?
As alluded to in the question, the primary (mostly) complete witnesses to the text of the Septuagint are codices bound up with the Christian New Testament. Pride of place goes to Vaticanus in which we have a nearly complete Greek OT. Sinaiticus and Alexandrinus also provide valuable witness. These codices have been dated to the 4th Century CE.
Unfortunately, the Great Uncials are far from perfect witnesses. As is well-known, revisions of the Old Greek translations probably began as soon as the translations had been made and culminated in the work of Origen in the 3rd C. CE. Origen's edition has been called "a monument to misguided industry" in recogntion of the challenge it introduces for modern text critical work.1 Although Origen himself used signs to mark changes in the received Greek text, these were often not preserved. Most subsequent Christian copies of the LXX are a mixture of the original Old Greek text and the editorial work of Origen. Manuscripts that pre-date Origen, then, are gold for LXX textual critics. This is all just to say: the OP's question is an important one!
The available manuscripts (all written on papyrus) fall into two groups. I have included here specific mention of some of the most significant along with brief notes about the nature of their import. They can be considered two groups:
Judean Desert: The most substantial group of papyri are fragments found among the so-called Dead Sea Scrolls. Although the majority of the DSS are Hebrew and Aramaic documents, a small fraction are in Greek. Among these, most are "documentary" rather than "literary"; that is, they record everyday transactions of a Greek-speaking community. However, a small collection of Greek texts, mostly from Caves 4 and 7, provide a witness to the Old Greek Bible.2
Lev. 26:2-16, late 2nd C. BCE. A small number of lexical discrepancies between the Leviticus manuscripts and later available LXX text has led Emanuel Tov to conclude that the 4QLXXLeva represents an earlier version of the Greek text, probably a freer translation of the Hebrew. Notably, John W. Wevers, in producing his Leviticus volume of the Göttingen edition of the Septuagint, did not select any of these readings.3
Lev. 2-5 with lacunae, 1st C. BCE or CE. These fragments are best known for their (to my knowledge) unique representation of the Hebrew Tetragrammaton using the Greek transliteration ΙΑΩ (iaō) rather than the more common translation (but see 2b., below) κυριος (lord).
c. 4QLXXNum, 4QLXXDeut, etc:
Portions (1-3 vv. each) of other Pentateuchal texts. These are fragmentary and of uncertain text critical value.
d. 8ḤevXII gr:
Portions of the Minor Prophets, 1st C. CE. Arguably the most important of the Greek scrolls from the Judean desert, this scroll includes remnants of 25 columns of the Greek Minor Prophets, together comprising more text than all of the Pentateuchal fragments above. This provides evidence (according to Barthélemy, who published pioneering work on the Scroll) for an intermediate stage of the text (termed the Kaige revision) which dramatically changed our understanding of the history of the Greek text.4
Other Papyri: These are mostly fragmentary bits of Pentateuchal texts, mostly found in Egypt. Among the most substantial:
Portions of Deuteronomy, 100 BCE. Known for its representation of the tetragrammaton using square Hebrew script.5
Approximately 20 scattered verses from Deuteronomy, c. 150 BCE. This is the oldest available manuscript of the Old Greek Bible. In addition to its antiquity, it is known for its blank spaces in place of the tetragrammaton, which have been the subject of much speculation.5
This is by no means a comprehensive compendium of witnesses, and the nature of the evidence is subject to change as further papyrological and textual evidence emerges, but hopefully, this has been enough to answer the OP's question
are there any scrolls and papyri or are there only codices? in the affirmative.6
1. John W. Wevers, as cited in Jobes, below, p. 53.
2. These are mostly not Qumran documents. Naḥal Ḥever and Jericho provide the largest fraction. For this and more: Emanuel Tov. The Greek Biblical Texts from the Judean Desert from Hebrew Bible, Greek Bible, and Qumran (Mohr Siedbeck 2008), Chapter 23.
3. This per Ulrich (The Septuagint Manuscripts from Qumran: a Reappraisal of their Value in Septuagint, Scrolls, and Cognate Writings [Manchester, 1990]), who offers specific analysis of each of the variants, many of which he (with Tov) considers likely original. A more detailed critique of Wever's decisions (and further support for the witness of 4QLXXLeva) can be found in Sarianna Metso and Eugene Ulrich's The Old Greek Translation of Leviticus in The Book of Leviticus: Composition and Reception Ed. Rolf Rendtorff and Robert Kugler (Brill, 2002).
4. Explaining that statement would, unfortunately, require a book -- one which I am not qualified to write, and one which would not fit here anyway. See either of the introductory texts below for an overview.
5. Both the Aramaic letters and the spaces are now widely (?) considered secondary. See also, a BH.SE answer about the representation of the tetragrammaton in the LXX.
6. With regard to the doublet
scrolls and papyri: while we have been discussing papyri (the type of material used), these documents were almost certainly also originally in the form of scrolls (the codex being a later invention), but because the manuscripts are mostly fragmentary, they are not patently scrolls.
In addition to Tov's work (n. 2), this answer was largely gleaned from:
Karen Jobes and Moises Silva. Invitation to the Septuagint Baker Academic, 2000. (Link is to the newer edition which I do not currently have access to in full.)
Jennifer M. Dines The Septuagint (T&T Clark, 2004).
Susan has given a typically well-informed answer to this question, and Noah has expanded on it with his link to Hurtado. Allow me to add one small word of caution: There is a wide-spread tendency of New-Testament scholars to date Bible manuscripts solely on the basis of their handwriting to slots of typically 50 years (e.g., "first half of the 2nd century CE"). Scholars working on "profane" Greek manuscripts of the same period do not as a rule accept that it is possible to date manuscripts so precisely merely on the basis of handwriting. There is a fundamental methodological problem here. In my own small area of professional expertise (Syriac and Arabic manuscripts) I would also be very hesitant to date any manuscript to a half-century slot unless it contained a colophon or some other clear-cut dating.
Larry Hurtado made a table of all known 2nd and 3rd century Christian manuscripts together with what form the manuscript takes. This includes manuscripts where it is unclear whether it is Christian or Jewish in origin and some definitely Jewish texts for comparison. I think that means it likely includes all LXX manuscripts. At any rate it gives a large number of partial LXX manuscripts that predate Vaticanus.
(The main purpose of this table is to provide evidence for Hurtado's assertion that early Christians had an unusual and strong preference for the codex over the book roll.)