I've always found the presence of a Jewish Turkic society in Eastern Europe to be really fascinating, especially because the modern-day countries of Ukraine and Russia would be vibrant centers of Jewish live about a millennium after the Khazars' heyday. While I understand that there is not any evidence continuity between these Khazars and later Eastern European Jewish populations, I find it difficult to believe that all traces of Khazar society vanished after the Rus' conquest. Societies generally do not vanish overnight, and other steppe peoples like the Cumans left a substantial cultural legacy in Eastern Europe after losing their own khanate. I would expect that the Khazars' Jewishness would have also inhibited assimilation into nearby non-Jewish societies.
I understand that to what extent Khazar society was Jewish is a matter of intense debate, but even if only the ruling class had converted to Judaism, I would think that their conversion would still leave a substantial impact on the cultural and religious landscape of Eastern Europe. Steppe societies were famous multicultural and religiously syncretic. Is there evidence of Jewish religious customs, names, or other cultural practices remaining in the region after Khazar society was destroyed?
The Radhanites connected communities across the breadth of Eurasia, so their travels would have likely brought news of the existence and ultimate demise of the Khazar monarchy to the Jewish communities of Germany and Persia. Western Khazar territories like Crimea were also not that distant from early medieval Jewish population centers like Lvov and Kiev. Is there any evidence of Khazar refugees seeking shelter within the early Ashkenazic populations of Eastern Europe, who would later come to settle former Khazar territory? I've read that Yiddish may suggest some Turkic influence, but provocative claims drawn on shaky evidence are not foreign to linguistics so I'm going to take that with a grain of salt. Wikipedia mentions Jewish warrior graves in Central Europe and the Balkans, but I would expect this has to do with raiding incursions rather than western Khazar migration.
Ashkenazi Jews in the 8th-10th centuries (the lifespan of Jewish Khazaria) were almost completely located in the Rhineland area of Western Germany and Eastern France. It wasn't until persecutions became markedly worse in the late 11th and early 12th centuries, revolving mainly around the Crusades, that the community really started their move eastward. When the general charter of liberties was instituted in Poland, the Khazars had been gone for 250 years at this point. I'm guessing anyone left would have probably converted to Islam or Orthodox Christianity and assimilated into either the Kieven Rus or other closely-related Turkic people like the Bulgars or Alans.
As for a legacy on Judaism, there is little evidence that they were anything other than the Rabbinic form of Judaism that already existed in the vast majority of the Jewish world (the Khazar Correspondence mentions study of the Mishnah and the Talmud). From what I've read, there are no real hints of Khazar religious scholarship which would have affected religious life outside their community. The rulers also seemingly adopted very common Jewish names. There were Khagans/Khagan Beks with names like David, Aaron, Joseph, Benjamin, and biblical king names like Hezekiah and Menasseh. However, the last recorded Khazar ruler was named Georgius, indicating that possibly a societal conversion to Christianity had taken place. Considering how long-term archaeologically-visible Jewish customs, largely burial practices, are fairly indistinguishable from Christian or Muslim ones, there is no real way to judge religions, save finding something like a golden crucifix buried a grave. One way that archaeologists in the Middle East attempted to discern the Israelites from other Canaanite people was a lack of pig bones, but even that wouldn't work because the Muslim people in region also didn't eat pork.