The Varangians would have originally spoken Eastern dialects of Old Norse of the sort that gave rise to today's Swedish, but they soon assimilated to the local East Slavic dialects that would give rise to modern Russian, Ukrainian, and Belarusian. They became known as the Rus, so we sometimes call this the Rus language or Rusian. The trade route would have gone through or at least were close enough to areas inhabited by people speaking Uralic languages, including the Finnic precursors of modern Finnish, Estonian, Karelian, and Veps; speakers of Old Permic, of which Udmurtian is a descendant; as well as groups known as the Volga Finns, precursors to the speakers of the modern languages of Mari, Erzya, and Moksha. There were also Balts like Latgalians and Lithuanians who spoke Baltic languages that are precursors of modern Latvian and Lithuanian. Further south we would have found Bulgars and Khazars, who mainly spoke diverse Turkic languages; Chuvash is thought to derive from the Bulgar language. These areas would have been diverse linguistically, with Ugric speakers (speaking languages related to modern Hungarian), Iranian speakers such as the Alans (speaking languages related to modern Ossetian), and probably speakers of North Caucasian languages as well.

Already there are several language families represented in this overview, including Indo-European (Germanic in the form of Old Norse, Slavic, Baltic, Iranian), Uralic (Finnic, Ugric), Turkic, and North Caucasian. To provide some bearings, I mentioned a lot of modern languages that would be related to the languages spoken in this period. As most of these languages were not written, we're forced to reconstruct the linguistic picture of a region based on our knowledge from later on in history. Not all the languages would have left modern descendants. Some might not even have left close relatives, and may have become extinct without a trace. In any case, over a millennium of language change separates all of these languages from their modern descendants if any; if you're familiar with the Old English of Beowulf, which was composed around the same time, it should give you an idea of how much languages can change over that time-scale.

I don't know about the lingua franca that would have been used on the Volga trade route, but to find out I would first see if there is any testimony from Arab and Persian travellers, geographers, and linguists like ibn Fadlan or al-Biruni. In late mediaeval times, Tatar, a Turkic language, would have been a lingua franca in the region (the Tatar language still thrives today in Tatarstan in the Russian Federation). But I have a feeling this is a later development due to the Turkic expansions and Mongol invasions, and the picture might have been different in the 9th or 10th century.

Edited because ibn Fadlan was Arab, though perhaps of Persian descent

There was a massive debate by the Volga Tatars over whether they are of Volga Bulgar descent.

The Volga Bulgars were a Muslim people who spoke a Turkic language related to the Chuvash branch of Turkic. But the modern day Chuvash people are non-Muslims.

The modern Volga Tatars are Muslims but they speak a Kipchak Turkic language. The Volga Tatars claim descent from the Bulgars and there are theories that during the Mongol conquests and Golden Horde rule, the modern Volga Tatar ethnicity experienced ethnogenesis when Kipchak speakers fused together with Bulgars under Mongol rule, with the new ethnicity adopting the Kipchak language and were known as Tatars by foreigners.

During the 19th century there was a Volga Tatar movement to reclaim Bulgar identity but some people noticed that the Bulgar language was related to Chuvash instead yet the Chuvash were not Muslims so there was an entire debate by academics over who the real heirs of the Bulgars were.