Sasuke would be extremely angry and determined to rescue Naruto. He would immediately start formulating a plan, using all of his skills and knowledge. He might even consider making some dangerous bargains or alliances if it means getting Naruto back.
His other friends like Shikamaru would use his intelligence to strategize. He would analyze Kaguya's previous actions and behaviors to predict where she might be hiding Naruto. And the rest of the Konoha 11 would band together, each contributing their unique skills, whether it's Ino's mind - transfer jutsu for intelligence gathering or Choji's strength in combat, all with the goal of rescuing Naruto.
His friends' reactions would be intense. Kakashi, as Naruto's teacher, would feel responsible. He would gather as much information as he could about the kidnapping. His friends from Team 7 and the other ninja in the village would band together. They would know that Naruto is strong, but also that the Kyuubi is a powerful and dangerous force. They would not rest until they found Naruto. Some might be angry at the Kyuubi, while others would be more focused on just getting Naruto back safely. They might also try to find ways to communicate with the Kyuubi, if possible, to negotiate for Naruto's release.
Another plot twist could be that Kaguya kidnaps Naruto to use his connection to the Nine - Tails. She might be trying to gain control over the Nine - Tails' power for herself. But Naruto, being the resilient character he is, manages to resist her control and instead starts to influence Kaguya. He might make her see the error of her ways or at least make her question her motives.
Sasuke would be furious. He already has a strong sense of justice and loyalty to Naruto. He'd likely immediately start looking for Naruto, using his Sharingan to track any possible leads. He might even go against the village rules if he thought the village was involved in Naruto's kidnapping.
Sakura would be worried sick. She might immediately inform Kakashi sensei and the rest of the village. Then, she would probably want to be part of the rescue mission. Sakura has grown stronger over time and would use her medical ninjutsu and combat skills to help save Naruto. She would also be constantly worried about Naruto's well - being and would encourage the others to move faster.
Yes, there are likely some fanfictions with this concept. Fanfiction writers often come up with unique and creative ideas, and the idea of Naruto having some sort of special connection or relationship with Kaguya Otsutsuki in a fan - created story is not unheard of. It could involve Naruto exploring Kaguya's past, or perhaps having a role in her redemption in the fanfic's alternate universe.
At first, Naruto would be stunned. But being Naruto, he'd quickly gather his wits. He'd probably joke around a bit to lighten the mood and try to get Hinata to open up. When he realizes she's serious, he'd start to analyze the situation. He'd think about what could have made Hinata do this and try to find a solution that would not only free him but also help Hinata if she's in some sort of trouble. He might also think about how this would affect their friends and the village as a whole.
Well, Naruto's character would be drastically different. Growing up with Kaguya, he might have a more aloof and powerful aura. Kaguya could teach him her unique jutsu from the start. He'd probably be more in touch with the Otsutsuki side of his powers. His sense of justice might still be there but expressed in a more cold - calculated way, like Kaguya. He'd also have a very different relationship with the other ninja in the village as he'd be seen as an outsider due to his unique upbringing.
His friends like Shikamaru would likely find the whole situation tiresome at first. He'd be thinking 'What a drag, now we have to deal with Naruto and his pacifier.' But then he might start to see the humor in it and use it as a way to tease Naruto in a friendly way. Choji would probably be more sympathetic and offer Naruto some snacks to go with his pacifier, seeing it as something that makes Naruto a bit more vulnerable and in need of comfort.
His friends from Team 7, like Kakashi, would feel responsible for Naruto's situation. Kakashi would likely use his resources to find a proper shelter for Naruto. Shikamaru and the others might come together to form a sort of support group for Naruto. They would pool their resources, like money and food, and also try to cheer him up by sharing stories and jokes.
The plot would change a great deal. For starters, the Akatsuki might not be as big of a threat to Naruto. Since he's being raised by Kaguya, he'd likely be much more powerful from the get - go. The relationship between Konoha and Naruto would be very different. Instead of being an outcast at first, he could be seen as a potential danger because of his connection to Kaguya. Also, the search for power and the various power - up arcs in the original story might not exist in the same way. Naruto would already have access to powerful Otsutsuki techniques.