Most likely Justus and Matthias; but the 120-member assembly's witness was the focus
"They" doesn't exist in the Greek...
Acts 1:24-26 (SBLG) emphasis added
24 καὶ προσευξάμενοι εἶπαν Σὺ κύριε καρδιογνῶστα πάντων, ἀνάδειξον ὃν ἐξελέξω, ἐκ τούτων τῶν δύο ἕνα,
25 λαβεῖν τὸν τόπον τῆς διακονίας ταύτης καὶ ἀποστολῆς, ἀφ’ ἧς παρέβη Ἰούδας πορευθῆναι εἰς τὸν τόπον τὸν ἴδιον.
26 καὶ ἔδωκαν κλήρους αὐτοῖς, καὶ ἔπεσεν ὁ κλῆρος ἐπὶ Μαθθίαν, καὶ συγκατεψηφίσθη μετὰ τῶν ἕνδεκα ἀποστόλων.
Acts 1:24-26 (NASB) emphasis added
24 And they prayed and said, “You, Lord, who know the hearts of all men, show which one of these two You have chosen
25 to occupy this ministry and apostleship from which Judas turned aside to go to his own place.”
26 And they drew lots for them, and the lot fell to Matthias; and he was added to the eleven apostles.
These verbs in vv24, 26 are where we get the English word "they" from "they prayed" (24) and "they drew" (26). We only have the conjugation of the verb being in third person plural. But, there is not subject of these verbs.
Therefore, the subject of these verbs is not important.
The meaning of the text is elsewhere...
We look at v24 because if that verb had a subject, this might be carried over to the subject of v26, but it doesn't. And, all three verses are sentences unto themselves. So, it is not like we can go back to the early part of a long, run-on sentence and find a subject as we might with Paul's writings.
The closest we can get is a subject and object from v15, when "Peter" (subject) spoke "in the midst of the brethren" (Dative/Locative, closest to an object of the speech we can get; the 120):
Acts 1:15 (SBLG)
Καὶ ἐν ταῖς ἡμέραις ταύταις ἀναστὰς Πέτρος ἐν μέσῳ τῶν ἀδελφῶν εἶπεν (ἦν τε ὄχλος ὀνομάτων ἐπὶ τὸ αὐτὸ ὡς ἑκατὸν εἴκοσι)·
Acts 1:15 (NASB)
At this time Peter stood up in the midst of the brethren (a gathering of about one hundred and twenty persons was there together), and said,
Acts 1:15 (my literal translation with word order [brackets denote noun case and verb person])
And in the days these stood Peter in midst [of] the brethren [he] said (and being at the same names who approximately one-hundred twenty)
Word order in Greek does not denote grammar as much because nouns have spelling case, but it denotes "importance". The day gets the first importance, Peter gets the second importance, the brethren gets third importance and longer, final, detailed elaboration, the public address ('he said') gets fourth importance.
V15 is almost like a legal record.
There were finalists in the lots drawn: Justus and Matthias
Don't forget who was put forth.
Acts 1:23 (SBLG) emphasis added
καὶ ἔστησαν δύο, Ἰωσὴφ τὸν καλούμενον Βαρσαββᾶν, ὃς ἐπεκλήθη Ἰοῦστος, καὶ Μαθθίαν.
Acts 1:23 (NASB) emphasis added
So they put forward two men, Joseph called Barsabbas (who was also called Justus), and Matthias.
Again, the English word "they" comes only from verb conjucation; there is no subject of this verb.
The closest thing we might get to a subject in vv24, 26 is these two men put forward in v23.
The crowd gets the emphasis
Everything done was done in witness of and in concert with the 120-head crowd.
The speech (vv15-22) was done before the crowd, who later agreed and acted.
The crowd witnessed the chusing of final candidates: Justus and Matthias (v23).
The prayer (vv24-25) was done with the crowd, in the crowd's presence, with the crowd's agreement—whether members of this crowd spoke is neither recorded and therefore nor important for us to know.
The casting/drawing of lots was done in witness of the crowd, in concert with the crowd, and formally validated by that crowd's witness.
This crowd is acting much like a legislative body. Interestingly, the word "church/assembly" ἐκκλησίᾳ [ekklaesia] (Strong's 1577) means "legislative body" in many uses. Here, Peter is not addressing ἐκκλησίᾳ [ekklaesia], but "brethren" ἀδελφός [adelphos] (Strong's 80). But, it's not in the meaning of the word describing the group, rather in the behavior of the group itself where we see the behavior of a legal, legislative oversight body, such as a committee or board or other voting body. Here, early in Acts, even before Pentecost, the "brethren" are behaving like a legal body, even without being called that.
Based on the text, how it went down:
The crowd assembled with the remaining 11 overseeing as a board of MCs (such as Peter's speech), with all things being done in agreement with the 11 and the 120.
During this, two finalists were put forth somehow, witnessed and in agreement.
Drawing of lots was most likely done between those two finalists themselves, probably like a rock-paper-scissors decision between two people. You wouldn't have someone else represent you in a choice such as this because its not helpful and overly complicated. The text doesn't specify because that, also, would be overly complicated.
In the end, Matthis was chosen at the witness and approval of the 120 and the other 11. This witness is the important part, making it valid.
In legal meeting minutes, it would seem that who actually cast these lots was a procedural technicality that was either obvious or inconsequential or both. Meeting minutes also wouldn't record which hand people drew the lots from—right or left—or which foot the enter the room with because that also doesn't really matter in a legal meeting record as this reads like.
We know the important part: there was a quorum to address specific business at a meeting, there were finalists, lots were cast, a result was validated by those present.