I believe the Republic of Karelia has sought greater autonomy within the Russian Federation since the collapse of the U.S.S.R., but that’s true of many regions within Russia, whether ethnically-based “republics” or mere Russian oblasts or krays. One thing to bear in mind is that “[a]fter a 1947 peace treaty between the Soviets and Finland confirmed the annexation, almost the entire Finnish population of western Karelia moved into Finland”; the population of the Republic of Karelia is now about 70% Russian (quotes and statistics from the Britannica article linked to above).
There doesn’t seem to be any evidence of any seething nationalist discontent in evidence on the official website of the Republic of Karelia. I admit I didn’t really go through that site in any great detail, as it seems a bit turgid, but the history of Karelia pages there don’t appear to talk about being oppressed by the wicked Russian imperialists or anything.
There may be more resentment from the Finnish side of things about their lost territories from the World War II era, but I don’t know if there is any particular constituency within what is now Russian territory to put any Finnish irredentism into effect.
My mother was born in a small town, Karppila, near Viipuri, which was the major port that the Soviets wanted during WW2. She arrived in the US in the late 1930s for a visit. She met Dad, the war broke out and she decided to stay. Her parents were still in Karjala (Karelia). When the Soviets invaded, Grandpa and Grandma fled, taking a pig and the sewing bag Mom had made in the first grade (they ate the pig but we still have the bag). They basically left by the back door as the soldiers came in through the front. They ran west. They built a new life on the west coast of Finland. It is my understanding that most Finns left Karjala.
I have often asked Mom if she wanted Karjala back. The answer was nearly always NO.
“But it’s your home.” “No, not anymore.”
“But it’s the birthplace of the Kalevala (a Finnish epic close to our hearts).” “That doesn’t change how I feel.”
I have wondered how much of her attitude came about because of her age. She was born in 1918, just one year after Finland’s independence from Russia. A couple of decades of independence and it was snatched away. During the collapse of the Soviet Union, she did once mention that she had hope, but after a few years, she returned to her old stance. It was during this time that I heard the most optimism about re-unification, but that seems to have died down considerably. It used to be very baffling to me, but I guess too much time has elasped since the Winter War of 1939.
From The Kalevala: While the young ones are standing round us,
Of the rising generation,
Let them learn the words of magic,
And recall our songs and legends,
Of the belt of Väinämöinen,
Of the forge of Ilmarinen,
And of Kaukomieli’s swordpoint,
And of Joukahainen’s crossbow:
Of the utmost bounds of Pohja,
And of Kalevala’s wide heathlands. - Elias Lönnrot
The short answer is that, from 1956 to the breakup of the Union of Soviet Socialist Republics in December of 1991, Karelia was an autonomous republic member of the Russian Soviet Federated Socialist Republic. It was a full republic of the USSR only during the period 1940 to 1956, from what I can tell; from 1923 to 1940 the part of Karelia known as Eastern Karelia, which had been established to be part of Russia by the treaty between Russia and Finland in 1920, was an autonomous republic member of the Russian Federation, the same status it had from 1956 to 1991.
Currently, it is a member of the Russian Federation, the official name of the country we call Russia.