Did Viking captives fight in Islamic armies as slave soldiers?

It's possible. That is, captives trafficked from viking raids might have reached the Islamic world and been recruited into slave units. It is, however, an extremely small outside chance until units of "Turkish" slave soldiers become important in the mid-800s. These slave soldiers were explicitly drawn from markets in Central Asia, and we know eastern trade goods (like dirham coins) made their way from the Caliphate to Scandinavia (and beyond) through these same markets. Written sources from the northern world, meanwhile, show a steady movement of viking captives from Western Europe toward Central Asia. The evidence seems to match up, especially for the 860s to 890s, as reports of viking raiding correspond to an intensification of Scandinavian trade with Central Asia. After that, evidence for viking slave raiding in Western Europe gets increasingly thin, even though evidence for exchange seems to spike.

It must be first admitted that our sources uniquely privilege Western Europe. There's no strong indication that medieval Muslim slaveholders thought much about where their slaves actually originated, as long as they were clearly outsiders. And from all the peoples living along the peripheries of the Abbasid Caliphate, Western Europeans have secured a privileged place in our understanding of the Middle Ages. We're at a point where we can say there was a sizable slave population in the Caliphate and some pretty clear evidence for slaving in Western Europe by a group connected to the Caliphate (i.e. vikings), but we're missing the stories for pretty much everyone else. It's as if we had really great sources for the interior of the Kongo but we knew pretty much nothing else about the transatlantic slave trade until slaves arrived at sugar plantations later in Jamaica. We'd get a pretty false idea about the overall picture. Similarly, the Viking-Age slave trade through Scandinavia was probably big news for people who participated, but it was probably just one flow of slaves among many people living in the Caliphate.

If it seems like I'm skirting the issue, I am. The sources—both textual and archaeological—suggest that it's possible that some Western Europeans ended up in Abbasid armies. But it's equally clear that any Western Europeans who did so failed to make their mark as Western Europeans in the elite Arabic literary culture of their day. Elite Arabic texts do speak vaguely of Turkish slaves, but we have more precise evidence for slaves arriving from Iberia (not via viking raiding), Byzantium, and East Africa. We also have a scattering of more mundane documents preserved almost entirely from Egypt, and these indicate a diverse pool of slaves drawn from various parts of Africa and the Eastern Mediterranean, without any reference (that I know of) to Turkish slaves or others who might have been trafficked by vikings. Slaves from northern markets probably got sucked into the major urban markets of Baghdad and Samarra but did not continue further south.

Abbasid units of slave soldiers probably drew from a diverse pool of recruits in these markets, and even though many were labeled as Turkish, we shouldn't read too much into that term. Contemporary writers were very loose in their use of ethnic terms for peoples encountered beyond the Caliphate's northern frontiers. Ethnic terms in late antiquity were often used to describe military units, and it doesn't seem like it often mattered where recruits came from as long as they came to act in the expected ethnic ways. (If we have Romans who adopted Germanic names, why not Franks who adopted Turkish ones?) And if the transatlantic slave trade is any measure, slave buyers often care more about the markets their slaves come from than their chattels' actual places of origin, which typically remained poorly understood and discussed only in terms of stereotypes. So the Turkish units here were probably either recruited from northern markets or expected to behave like people recruited from those markets. Some captives trafficked by vikings might have ended among their ranks, but if so, nobody left an enduring record of their presence.

The phrasing of the question also suggests that OP might be interested in whether any captured vikings could have become slave soldiers to the Abbasid caliphs. That answer is no. Troops under Islamic leaders might have captured some viking raiders in Iberia, but these rulers were not subservient to the Abbasid caliphs, and it does not seem like slave units were used at that time by non-Abbasid Muslim elites. If vikings captured by Western Mediterranean rulers were enslaved, they would have been put to other uses.

Finally, to poke an important hole in Fynn-Paul's argument, it's significant that most people living under Muslim rulers were not actually Muslim until as late as perhaps 1200. That is to say, Muslims had access to plenty of slaves from the peoples already living in their communities, since these were already slave societies when the Islamic conquests swept through in the 600s. Fynn-Paul obscures this by referring to Islam and its demand for slaves, rather than digging into how these societies functioned and whether it's even accurate to talk about such things as an "Islamic" no-slaving zone or an "Islamic" demand for slaves. I'm not sure Fynn-Paul has given rigorous thought to how Islam itself (i.e. the religion) could have been connected to enslavement and the slave trade during the Abbasid period. So his article seems elegant in its outlines, but it breaks down when you ask who was doing what. Then its gross abstractions become problematic. In this case, I would suggest that a more rigorous use of primary sources might have helped a reconsideration of many underlying premises before this article went to press.