A vengeful villain whose plans were once thwarted by a hero could be the kidnapper. He might think that by kidnapping Midoriya, he can draw out the hero and get his revenge. Midoriya is just an innocent pawn in his evil scheme.
There's a possibility that it's a secret organization that has been observing Midoriya from afar. They think he has some sort of latent power that could be harnessed for their own purposes, so they kidnap him to try and unlock this power.
In a 'midoriya kidnapped as a child fanfiction', it could start with Midoriya being lured away by a seemingly friendly stranger who turns out to be a kidnapper. The kidnapper might be working for a powerful enemy of the heroes. Midoriya, being a smart and resourceful kid, tries to leave clues for his rescuers. His mother, Inko, is distraught and contacts the heroes. The heroes then have to search through different areas, following the sparse clues Midoriya has left. There could be a lot of emotional turmoil among the characters as they worry about Midoriya's safety and try to find him as quickly as possible.
It could be an old enemy of Keith's, someone from his past who has a vendetta against him. Maybe a former rival from his days in the military academy who has now turned to more devious means.
In some creative fanfiction, it could be an alternate version of Spiderman himself from a different dimension. This Spiderman has gone bad and kidnaps a fan from our dimension as part of some grand, dark plan that involves crossing dimensions and causing chaos.
A power - hungry nobleman from a neighboring land could be the kidnapper. He might believe that by having Rapunzel in his possession, he can gain some sort of political advantage or use her as a bargaining chip against her kingdom.
It could be Orochimaru. He's always been interested in obtaining powerful individuals for his experiments, and Naruko with her unique abilities would be a prime target.
It could be a new villain that has emerged, someone who is jealous of Gohan's power and wants to make it his own. This villain might be from a far - off planet and has heard of the Saiyan legend.
It could be a group of rogue Decepticons who are acting on their own. They might not follow Megatron's orders but still see value in kidnapping Arcee. Maybe they have a personal vendetta against her or think they can get rich by selling her to the highest bidder in the Cybertronian underworld.
Maybe it's a group of villains working together. They could be from different factions but united by the idea that Izuku has the potential to become a huge threat to their plans. They might kidnap him to either eliminate him or turn him to their side. Also, it could be a lone, obsessed fan of a villain who wants to prove themselves by bringing Izuku to their idol.
A group of outcasts from different islands might kidnap Moana. They feel that they have been wronged by the 'normal' society that Moana represents. They hope that by kidnapping her, they can get the attention they deserve and force the islands to change their ways. But Moana listens to their grievances and tries to find a way to help them while also finding a way home.
It could be Zeref. Since Zeref has an interest in powerful magic users and Natsu is a strong dragon slayer, Zeref might kidnap Natsu to study his magic or use him in some dark plan related to his quest for immortality or power.