The writer of Hebrews demonstrates an extensive knowledge of the Old Testament and the LXX in particular.
In Hebrews 11:37 we read ἐλιθάσθησαν, ἐπρίσθησαν, ἐν φόνῳ μαχαίρης ἀπέθανον, περιῆλθον ἐν μηλωταῖς (sheep skin), ἐν αἰγείοις δέρμασιν, ὑστερούμενοι, θλιβόμενοι, κακουχούμενοι.
The LXX has five references to the same same word all in relation to Elijah
1 Kings 19:13 καὶ ἐγένετο ὡς ἤκουσεν Ηλιου καὶ ἐπεκάλυψεν τὸ πρόσωπον αὐτοῦ ἐν τῇ μηλωτῇ ἑαυτοῦ καὶ ἐξῆλθεν καὶ ἔστη ὑπὸ τὸ σπήλαιον καὶ ἰδοὺ πρὸς αὐτὸν φωνὴ καὶ εἶπεν τί σὺ ἐνταῦθα Ηλιου
Kings 19:19 καὶ ἀπῆλθεν ἐκεῖθεν καὶ εὑρίσκει τὸν Ελισαιε υἱὸν Σαφατ καὶ αὐτὸς ἠροτρία ἐν βουσίν δώδεκα ζεύγη βοῶν ἐνώπιον αὐτοῦ καὶ αὐτὸς ἐν τοῖς δώδεκα καὶ ἐπῆλθεν ἐπ᾽ αὐτὸν καὶ ἐπέρριψε τὴν μηλωτὴν αὐτοῦ ἐπ᾽ αὐτόν
Kings 2:8 καὶ ἔλαβεν Ηλιου τὴν μηλωτὴν αὐτοῦ καὶ εἵλησεν καὶ ἐπάταξεν τὸ ὕδωρ καὶ διῃρέθη τὸ ὕδωρ ἔνθα καὶ ἔνθα καὶ διέβησαν ἀμφότεροι ἐν ἐρήμῳ
2 Kings 2:13 καὶ ὕψωσεν τὴν μηλωτὴν Ηλιου ἣ ἔπεσεν ἐπάνωθεν Ελισαιε καὶ ἐπέστρεψεν Ελισαιε καὶ ἔστη ἐπὶ τοῦ χείλους τοῦ Ιορδάνου
2 Kings 2:14 καὶ ἔλαβεν τὴν μηλωτὴν Ηλιου ἣ ἔπεσεν ἐπάνωθεν αὐτοῦ καὶ ἐπάταξεν τὸ ὕδωρ καὶ οὐ διέστη καὶ εἶπεν ποῦ ὁ θεὸς Ηλιου αφφω καὶ ἐπάταξεν τὰ ὕδατα καὶ διερράγησαν ἔνθα καὶ ἔνθα καὶ διέβη Ελισαιε
In all five cases our English translations based on the Hebrew text read 'mantle' or 'cloak' The Hebrew in 1 Kings 19:13 for example is אַדַּרְתּ֔
†אַדֶּ֫רֶת S155 TWOT28c GK168 n.f. glory, cloak—א׳ abs. Jos 7:24;
אַדָּ֑רֶת Ez 17:8; cstr. Gn 25:25 + 4times; אַדַּרְתּוֹ 1 K 19:13 + 3
times; אַדַּרתָּם Zc 11:3—1. glory, magnificence, of vine Ez 17:8 (so
Thes MV, but < adj. f. from אַדִּיר q.v.), of shepherds Zc 11:3 (or
sub 2). 2. mantle, cloak (wide garment) of hair שֵׂעָר Gn 25:25 Zc
13:4 (as proph. mantle, so perhaps 11:3 of shepherds = false proph.)
cf. of Elijah 1 K 19:13, 19 2 K 2:8, 13, 14; but אַדֶּרֶת שִׁנְעָר
טוֹבָה (fine mantle of Shinar = Babylonian mantle—doubtless costly)
Jos 7:21, 24 (J) & (late) א׳ alone Jon 3:6.
[Brown, F., Driver, S. R., & Briggs, C. A. (2000). Enhanced Brown-Driver-Briggs Hebrew and English Lexicon (electronic ed., p. 12). Oak Harbor, WA: Logos Research Systems.]
This is why we don't find reference to sheepskin in our Old Testament but the author of Hebrews did in his.
The reference to goatskin is a little more obscure. The term αἰγείοις δέρμασιν (goat skin) as found in Heb 11:37 isn't found anywhere in the LXX.
Zechariah 13:4 reads "καὶ ἔσται ἐν τῇ ἡμέρᾳ ἐκείνῃ καταισχυνθήσονται οἱ προφῆται ἕκαστος ἐκ τῆς ὁράσεως αὐτοῦ ἐν τῷ προφητεύειν αὐτόν καὶ ἐνδύσονται δέρριν τριχίνην ἀνθ᾽ ὧν ἐψεύσαντο"
δέρριν translates as a covering made of skin/ leather clothing, see for example Lust-Eynikel-Hauspie which contains this entry:
Ex 26,7 cloack (of skin) Zech 13,4; (tent) curtain (of goats'
hair) Ex 26,7; curtain (of skin or hide) Jgs(A )4,18
τριχίνην translates as 'made of hair' or Hairy see for example Thayer or Louw-Nida
It is possible that the writer of Hebrews is alluding to his reading Zech 13:4 as some commentators suggest, eg Jamieson, Fausset, & Brown
as the “goat-skins” were black (compare Zec 13:4)
Many bible dictionaries identify 'sackcloth' as being goats hair, see for example:
SACKCLOTH. A coarse cloth (Heb. śaq, Gk. sakkos, from which the Eng.
word is derived), usually made of goats’ hair (Siphra 53b) and black
in colour (Rev. 6:12).
[Thompson, J. A. (1996). Sackcloth. In D. R. W. Wood, I. H. Marshall, A. R. Millard, J. I. Packer, & D. J. Wiseman (Eds.), New Bible dictionary (3rd ed., p. 1032). Leicester, England; Downers Grove, IL: InterVarsity Press.]
However this option seems less likely as the writer does not use the term (σάκκος – sack cloth) but αἰγείοις δέρμασιν. The writer's familiarity with the LXX would include a familiarity with the term σάκκος - see for example Gen 37:34 and Lev 11:32
Whilst the question does not relate to the term "sawn in two" which is also found in Hebrews 11:37 the fact that the author uses it might be helpful in answering the question. There is a Jewish apocryphal book called ‘The Ascension of Isaiah’, and in chapter 5 of that book we read there the tradition that Isaiah was killed by being sawn in two.
The Ascension of Isaiah
Ch 5:2 And when Isaiah was being sawn in sunder, Belchira stood up,
accusing him, and all the false prophets stood up, laughing and
rejoicing because of Isaiah.
Ch 5: 11- 14 And they seized and sawed in sunder Isaiah, the son of
Amoz, with a wooden saw.And Manasseh and Belchira and the false
prophets and the princes and the people [and] all stood looking on.
And to the prophets who were with him he said before he had been sawn
in sunder: "Go ye to the region of Tyre and Sidon; for for me only
hath God mingled the cup." And when Isaiah was being sawn in sunder,
he neither cried aloud nor wept, but his lips spake with the Holy
Spirit until he was sawn in twain.
This reference to tradition points us to understand that the author of Hebrews was not just familiar with the Old Testament canon but with Jewish tradition as well. The author could be referring to Jewish traditions.
The reference to sheepskin is an reference to Elijah mantle/ cloak.
The reference to goatskin is not easy to tie down to any specific Old Testament text, it could be an allusion to Zech 13:4, but equally it could be being drawn from Jewish tradition or be a reference to sackcloth.