As per @robev's point in the comments it is brought down by Rav Moshe Sternbuch in his teshuvos v'hanhagos on Tisha B'Av.
The relevant excerpt mentioning the Vilna Gaon is in the first paragraph and translates as follows:
Russia's recent might is a sign of the coming Moshiach
Behold recently we have seen the might of Russia in the world who conquered half of Crimea. And it is accepted by the Gra (the Vilna Gaon) that, "When Russia become mighty that the impurity becomes mighty/increases." And the Gra reasons the klippah (literally "shell" - see here) of Russia is "Dubiel" (refer to Yoma 77a) that is inherited from the klippa of Persia, and since they are also Greek Christians it comes out that they also have mixed in with them the klippos of both Greece along with Edom.
The paragraphs that follow develop this idea...
And similarly, the author of "Binyan Olam", Rav Yitzchak Isaac Chaver a student of Rav Chaim of Volozhin zt"l said before his death to his student Rav Dovid Luria zt"l that, "If Russia conquers all the city of Crimea (and specifically the Kerch fortress) one can already expect the onset of Moshiach.
And likewise, HaGaon Rav Shlomo Zalman Riger zt"l, the av beis din of Brisk said that it is accepted by him from the disciples of the Gra in the name of the Gra that, "The kingdom of Russia will strengthen before the coming of Moshiach and the nations of the world will be scared of them."
And it is an accepted fact that HaGaon Rav Yitzchak the son of Rav Chaim of Volozhin zt"l was worried of war from the Russians and the son of his sister, Rebbe Yitzchak of Kovno, sent to the Saba Kaddisha of Ruzhin zt"l to ask him to act by the ties he had with the heads of Russia (the Chief Admiral in the Black Sea and two other Navy builders in the Black Sea) to listen out for the situation with the war of Russia, and the Saba Kaddisha answered him, "I agree that Russia is a bad klippah" and he added further that his elder, Rebbe Avraham HaMalach completely hated Russia and even held that the increasing might of Russia is a sigh that (the emergence of) Moshiach was getting stronger....