It's most certainly your translation that is a bit free.
In the original, the passage you are referring to talks about "gullnar tǫflur", 'golden tables' (stanza 58 in Codex Regius, 54 in Hauksbok; this line is the same in both, despite some other minor variations in the stanza). These are likely the same tables that are mentioned in stanza 8, where the idyllic life of the Aesir in the new creation is outlined.
While Chess seems to have reached Europe from the Middle east during the Viking age, and the Vikings are recorded as having direct contact with the Muslim world, there is a better candidate in a set of games the Norse played, known as Tafl or Hnefatafl, another two-player strategic game without any luck component. It could also be some other game that we do not know as much about.
Finally, while Snorri would almost certainly have known about chess, he had nothing to do with writing the Völuspá; his Edda is an instruction book in poetry and mythology, which sometimes quotes from older works. Völuspá is such an older work, included in the medieval compilations that later were standardised and given the same title that Snorri had used for his work.
Here is Sophus Bugges edition of the Völuspá, containing parallel text from three different sources. Blackwell's translation from 1906, from project Gutenberg, might be dated, but one can see that he uses "tables" as the translation.