As the OP correctly notes, Hebrews 1:6:
ὅταν δὲ πάλιν εἰσαγάγῃ τὸν πρωτότοκον εἰς τὴν οἰκουμένην, λέγει Καὶ προσκυνησάτωσαν αὐτῷ πάντες ἄγγελοι θεοῦ. (Westcott and Hort)
And again, when he brings the firstborn into the world, he says, “Let all God's angels worship him.” (ESV)
is most likely a quote of an LXX version of Deuteronomy 32:43
εὐφράνθητε, οὐρανοί, ἅμα αὐτῷ, καὶ προσκυνησάτωσαν αὐτῷ πάντες ἄγγελοι Θεοῦ1 εὐφράνθητε, ἔθνη μετὰ τοῦ λαοῦ αὐτοῦ, καὶ ἐνισχυσάτωσαν αὐτῷ πάντες υἱοὶ Θεοῦ· ὅτι τὸ αἷμα τῶν υἱῶν αὐτοῦ ἐκδικᾶται, καὶ ἐκδικήσει καὶ ἀνταποδώσει δίκην τοῖς ἐχθροῖς καὶ τοῖς μισοῦσιν ἀνταποδώσει, καὶ ἐκκαθαριεῖ Κύριος τὴν γῆν τοῦ λαοῦ αὐτοῦ.
Rejoice, ye heavens, with him, and let all the angels of God worship him; rejoice ye Gentiles, with his people, and let all the sons of God strengthen themselves in him; for he will avenge the blood of his sons, and he will render vengeance, and recompense justice to his enemies, and will reward them that hate him; and the Lord shall purge the land of his people. (Brenton)
Psalms 97:7 is similar,
αἰσχυνθήτωσαν πάντες οἱ προσκυνοῦντες τοῖς γλυπτοῖς οἱ ἐγκαυχώμενοι ἐν τοῖς εἰδώλοις αὐτῶν· προσκυνήσατε αὐτῷ πάντες οἱ ἄγγελοι αὐτοῦ
Let all that worship graven images be ashamed, who boast of their idols; worship him, all ye his angels.
but has several differences and thus is comparatively unlikely.
Dead Sea Scrolls
On the heart of the question, BibleQuery.org states that a DSS version (4QDeut) of "Deuteronomy 32:43 has 'let all God’s angels worship him' while the Masoretic text does not." The New American Commentary agrees. Likewise, the NLT translation notes and ESV translation notes show that the DSS generally agrees with the LXX against the MT on the parts of this verse.2
Analyzing these difference, Michael Heiser writes:3
It is significant that the Masoretic text lacks a second line in
what should be the first pairing. Even more striking is the fact that
this missing colon is the one in which reference is made to divine
beings in the Qumran and Septuagintal texts. In these latter two
texts each colon has its partner. This argues strongly that the Masoretic
text originally had a bicolon, a pairing that was deliberately
eliminated to avoid the reference to other "divine beings."
... those who defend the priority of the Masoretic text would have to argue
for accidental changes in Deuteronomy 32:8 and
in 32:43—changes that produced false readings in favor of angelic
beings in both cases, while simultaneously accounting for all the
consonants in 4QDeut. Such a coincidence is possible, but
it stretches credulity to argue that the Masoretic text of Deuteronomy
32:8 and 43 best represents the original text... It is far more likely that both texts were intentionally altered in the Masoretic text for the same reason, namely, to eliminate a reference to heavenly beings in order to avoid allegedly polytheistic language.... Therefore the reading in the Septuagint sufficiently explains how the Masoretic reading could have arisen, but the alternative does not.
In other words, the MT is likely a corruption of the original. And the missing words were likely dropped in an overzealous attempt to eliminate polytheistic language. The DSS, LXX, and Hebrews 1:6 thus represent the more original reading.
Also of note is that the LXX reading is supported by The Book of Odes, a work which contains excerpts from several books of the Bible, collected for use in liturgical songs. (The work is considered canonical by the Eastern Orthodox Church.) Chapter two of Odes corresponds to chapter 32 of Deuteronomy. Odes :43 supports the "angels" reading found in Hebrews and some, but not all copies the of LXX. The work probably does not predate Hebrews and thus is unlike be the source of the Hebrews quote. It thus provides an independent witness to the validity of this reading. This reading has convinced textual critics such as Tim McLay that other versions (4QDeut, MT) derive from a text very similar to the Old Greek (critical reconstructed original Septuagint text), a text that closely matches the Hebrews 1:6 wording.4
1 There is a textual variation here. Some copies read υἱοὶ Θεοῦ "sons of God" instead of ἄγγελοι Θεοῦ "angels of God". In the OT, "sons of God" is generally a synonym for angels; thus the underlying Hebrew would likely be the same.
2 The DSS version lacks the 3rd and 4th clauses of the LXX. However, our quote is the second clause. The MT contains only the first clause.
3 Deuteronomy 32:8 and the Sons of God by Michael Heiser
4 The Use of the Septuagint in New Testament Research by Tim McLay