I found this story in Hungarian translation in this book, collected and translated by Erdődi József in 1960. As far as I can tell, the book did not give a source, and I could not trace the story to any English publication. If anyone has anything to add, please let me know! I would love to find out more.
(The translation is mine. Here is the text online in Hungarian)
Once upon a time there was a large, dark country by the sea. In that country, the sun never shone, its people never saw sunlight. The land was covered in a dark cloud that created such darkness that the people could barely see each other. They called this land the Land of Darkness. It's inhabitants were called hut-dwellers, because they had no houses, only rickety huts. The huts were woven from switches, covered in bark and moss, and the cruel wind could go through and through them. The people of the Land of Darkness lived in great misery.
In the middle of the Land of Darkness, however, there was a round mountain. On the round mountain there were many, many trees, a whole forest. In the middle of the forest there was a large log house. Inside the house it was warm, and there was an abundance of things. In its rooms lived seventy shadowy siblings. They were the only ones with a log house in the Land of Darkness. Around the house there was a wooden fence, and inside it a hundred thousand reindeer. The shadowy siblings did not use the reindeer for anything, but they also did not give any of them to the hut-dwellers. This is how things went on for a thousand years, then another thousand, then a third thousand. The hut-dwellers thought it would be like this forever, unless the winter went away.
One day - no one knows when - a tall men arrived to the Land of Darkness, riding a beautiful reindeer. His beard reached his knees, and when he spoke, his eyes shone bright enough that his face could be seen even in the dark. The hut-dwellers saw how handsome and beautiful he was, and how intelligent his gaze was. When he spoke, they all listened carefully.
"My friends, you all live in darkness, because you don't know the Sun. But the Sun exists, even if none of you have seen it. If you find the Sun, this land will also be warm and bright."
"What is a Sun?", the people wondered. They had never heard about it before. The shadowy siblings also heard the stranger speak. They heard it, got angry, and started berating the people.
"Stupid hut-dwellers! Why are you listening to this stranger's nonsense? How could something exist if no one has seen it before? He is only here to rile you up! Listening to him is a mistake. He deserves a beating for telling you made-up tales!"
The hut-dwellers thought and wondered. Maybe the shadowy siblings were right? Maybe it would be better to kill the stranger? The bearded wise man just watched them and shook his head. The light went out of his eyes. He turned his reindeer and rode away, disappearing as if he'd never been there. They could only hear his voice in the dark:
"From this day on I will only appear to those who believe in the existence of the Sun."
There was a boy among the hut-dwellers. He was poor like all the others, but he never humbled himself by going begging to the shadowy siblings. He was proud and strong. Time passed, and everyone forgot the wise old man, except for him. He went to the place where the reindeer moss grew, looked up at the black sky, and said to himself:
"I believe that the Sun exists. But how can I find the bearded old man?"
The moment he said these words, the moss and lichen parted, and a beautiful reindeer appeared in front of the boy.
"Get on my back," it said. The boy mounted the reindeer, and it galloped with him across the moss, the swamps, the black lakes. Suddenly, it stopped in front of a granite rock. On the rock sat the old man with the long beard.
"Welcome!," he greeted the boy "I knew there was someone among the hut-dwellers who would come find me. You are a good man, you will go far, my son."
"Thank you for your kind words. But tell me, where can I find a piece of the Sun?"
"You will have to work hard to earn the Sun. You will have to weave a basket. Ask for a single hair from every hut-dweller, and use those to make your basket."
The boy returned to his people. He talked to everyone, and convinced each of them to give him one hair. Once he'd collected all, he began to weave a basket. He worked on the tiny basket for seventy days and seventy nights. By the time he finished, his strength and wisdom grew. Then he returned to the moss and lichen field, looked up at the black sky, and said to himself:
"The basket is done. How could I get a piece of the Sun?"
The moment he said it, the moss and lichen parted, and the beautiful reindeer appeared.
"Get on my back," it said, and galloped with the boy across the moss, the swamps, the black lakes. They had a long journey, until suddenly they saw red light. The wise young man saw the great red Sun on the edge of the horizon.
"Are you not afraid of fire?," asked the reindeer.
"I am not afraid of anything," the wise boy responded.
"Then open the basket, but hold it firmly, and brace yourself well!"
The wise boy did so. The reindeer rode at the sun. It ran at the Sun, stabbed it with its soft antlers, and a piece broke off the Sun, falling into the boy's basket. Then they turned around, and rode back. The moment they arrived to the land of the hut-dwellers, the magical reindeer disappeared.
The boy stood in front of the hut-dwellers, and said:
"You all gave me one hair each. I wove a basket from your hair, and brought you a piece of the Sun. Let's let it out of the basket. Let it brighten the sky!"
The moment he said it, the shadowy siblings were already riding down the mountain. They shook their fists and yelled: "Don't you dare! Don't you dare let out the Sun! The lakes will dry up! The iron will melt in the ground and flood our houses! You will go blind and we will all burn!"
The shadowy brothers surrounded the wise boy, trying to tear the basket from his hands. But the hut-dwellers rose up to defend him.
"Don't touch him!", they yelled. "We will not give you the basket!"
The shadowy siblings grew enrages. They grabbed the wise boy and dragged him towards the swamp. They wanted to drown him in the swamp, and throw the basket after him. But the hut-dwellers grew angry too. They grew bold, and for the first time in their life, they grabbed rocks from the ground. They threw the rocks at the shadowy siblings. The rocks rained down on them, and the shadowy siblings drew sharp fish bones from under their clothes, and started stabbing the hut-dwellers. But they resisted. Blood was spilled, and a battle began. In the middle of it, suddenly the basket sprang open, and the first ray of sunshine broke out. The sky turned red with light, the swamps bathed in the colors of dawn. The shadowy brothers all burned, their ashes falling into the swamp. The wise boy was standing there, looking at the sky, the bright rays of sunlight. All the hut-dwellers marveled at the sky. The water of the lakes turned blue, the mosses and lichens gained bright colors - white, red, yellow, and green. A miracle happened in the Land of Darkness.
"Who are you, wise man?," the hut-dwellers yelled "Who are you, who brought us this miracle?" They all ran to him and asked: "Wise boy! Now we know and can see that the Sun exists. But this is only a small piece of it. How can we get the whole Sun?"
The moment they asked, the mosses parted, the magic reindeer appeared, and said to the wise boy:
"Tell them to herd all the reindeer here from behind the fence of the shadowy siblings. The boy told the hut-dwellers:
"Go to the house of the shadowy siblings. Break down the fence, and bring the reindeer here. They belong to you now."
The hut-dwellers broke down the fence, and herded the hundred thousand reindeer of the mountain. They mounted them and rode away. They rode until they reached the Sun.
"Are you not afraid of fire?", asked the wise boy.
"We are not afraid of anything. Tell us how to get the Sun."
"Ride towards the Sun with an open heart, and take a ray of sunshine each into your hearts."
The hut-dwellers opened their hearts, and rode fast at the Sun. Each of them took a ray of sunshine into their heart. A hundred thousand hearts warmed up.
"Now, line up the reindeer!"
They did. The magic reindeer poked at the sun with its antler, the Sun slid off the sky, and rested on the back of the animals. The hundred thousand hut-dwellers set out on the hundred thousand reindeer, all balancing the Sun on their antlers carefully.
Ever since then, the Sun has been shining bright above the tundra. The lakes are blue, the hut-dwellers fish in their clear waters. The swamps dried out, and they have been replaced by colorful wildflowers and soft grass. Endless forests whisper along the sea.
The wise man still lives, and he will never die, because he is the one who brought the Sun to the people of the Land of Darkness.
(Note: I translated "black siblings" as "shadowy siblings" to avoid connotations of race, and also because it is closer to the Hungarian meaning.)
