Yes, there were definitely non-Muslim intermediaries in the Black Sea slave trade! However, the identity (and indeed, the participation levels) of these intermediaries shifted a great deal over time. Prior to the 1470s, for example, the Genoese were major players in the Black Sea slave trade—though it should be noted that (1) their main Muslim partners here were the Mamluks rather than the Ottomans, and (2) that they also sold slaves to Christians. (At times the Venetians overtook the Genoese as the chief merchants in the Black Sea slave trade, but the point more broadly is about the presence of Italian Catholics as middlemen.) Hannah Barker has written extensively about the Italian slavers in the Black Sea, and I'd recommend her That most precious merchandise: the Mediterranean trade in Black Sea slaves, 1260-1500 (U. Pennsylvania Press, 2019) as an introduction to the subject.

Italians were not the only merchants involved in the slave trade, as Barker's investigation of the mid-fourteenth-century market at Tana makes clear:

All of the slave buyers were Italian men, but the sellers were diverse. Nineteen percent were non-Italian residents of Tana, including Abduracoman (ʿAbd al-Raḥmān) the Saracen and Ascelan the Armenian, who each sold four slaves. Six sellers were women, including the Tatar widow Erda, two Saracen widows named Adia, an Armenian widow named Chatlumelich, a Russian widow named Ochinolato, and Katerina of Sevastopol. There were also sellers from Caffa and Pera as well as Porto Pisano and the Golden Horde (including Gazaria and Russia as well as the area around Tana). Finally, a single Venetian named Domenicus of Florentia sold forty-three slaves during September-October 1359 and May-September 1360, appearing and disappearing in sync with the Venetian galleys.

In 1475, however, the Ottomans and their Crimean vassals conquered the Genoese trading colonies at Cembalo (modern Balaklava), Soudaia (Sudak), and Caffa (Feodosiya). This spelled the end of the Genoese presence in the Black Sea—Italians would no longer be major players in the Ottoman slave economy, though small numbers of Italians would continue to participate in the slave trade in Black Sea ports. At the same time, we see the rise of the Crimean Khanate as a major slaving power, though one more involved in procuring slaves (through trade, but more often as plunder) than in transporting them to the points of final sale. Greek, Jewish, and Armenian merchants were the most prominent non-Muslim intermediaries in this period of the trade.

By the seventeenth century, this dynamic may have begun to shift again. A mid-seventeenth-century Ottoman list of slave dealers (in Alan Fisher, "The Sale of Slaves in the Ottoman Empire") contains entirely Muslim names. In theory, various fetvas and laws proclaimed by religious authorities prohibited non-Muslims—or even non-Sunni Muslims—from owning slaves. The overwhelmingly Muslim makeup of the slavers may have been a result of official pressure in this area, though technically speaking the bans tended to be on the sale of slaves within the Ottoman Empire to members of these groups. A 1609 edict of the kadı of Constantinople (also in Fisher) reiterates this position, while also shedding light on the level of participation in the slave trade across all levels of Constantinopolitan Muslim society:

It has been learned that slave dealers have come to the Sultan's court city (Istanbul) and have been selling slaves to Jews and Christians, have been selling slaves to the ambassador of the Şah of Iran, to Jews, Christians and Kızılbaş in the Bedistan; some of these slave dealers have been women; even some members of the class of Sipahi and Janissary have been dealing in slaves and sending these slaves to other countries; it is important that the security officials stop all of these practices and evict such unseemly persons from the slave market. It is prohibited that anyone from either Sipahi or Janissary classes and other non-registered dealers be allowed to sell in the slave market. And further, it is prohibited that any slaves be sold by anyone to Jews, Christians or Kızılbaş.

Outside of Ottoman cities, however, Greeks, Jews, and Armenians continued to play an active role in the trade of slaves to the Ottoman Empire. The Polish border authorities repeatedly arrested Jewish and Armenian merchants carrying children who were kidnapped or bought from Polish families; travelers noted the prevalence of all three non-Muslim groups in the Circassian slave trade; and so on. Despite the presence of non-Muslim intermediaries, however, by this point the slave trade was not the sole province of non-Muslims: Turkish and Tatar merchants, as well as converts, played a significant role in the trade. But as the example of Abduracoman the fourteenth-century Saracen reminds us, this had been the case—at least in part—for a long time. Ottoman slavery, and the Black Sea trade especially, is a difficult phenomenon to lay at the door of any one ethnic or religious group.