The kidnapper's motive could be revenge against Kaiba. By kidnapping Tea, they know they can draw Kaiba out and make him suffer. Maybe Kaiba defeated this person in a duel before and this is their way of getting back at him.
Since I don't know the details of the 'kaiba saves a kidnapped tea' fanfiction, it's hard to give a precise summary. However, it could start with Tea being kidnapped by some unknown enemy, perhaps someone who wants to use her to get to Kaiba or has a grudge against her. Kaiba then discovers her disappearance and embarks on a dangerous journey to find her. Along the way, he may encounter allies or more foes, and there could be a lot of suspense and action as he tries to free Tea from her captors.
Another motive could be revenge against the girl's family. Maybe the kidnapper has a grudge against her parents or relatives. Gagging the girl is a way to keep her quiet while they carry out their plan of getting back at the family in some way.
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.
It could be a jealous rival from his own village. Maybe someone who is envious of Shikadai's natural talents and popularity, so they decide to kidnap him to get him out of the way.
It could be an envious rival. Maybe someone who's always been in the shadow of Osomatsu and his brothers and decides to take matters into their own hands by kidnapping Osomatsu. Another option is a deranged collector. This person might see Osomatsu as a rare item to add to their collection of oddities. Or it could be a vengeful spirit. In a more supernatural setting, a spirit with a grudge against the Matsuno family could kidnap Osomatsu as a form of revenge.
It could be a rogue werewolf. Maybe this werewolf has a grudge against Stiles' pack and thinks that by kidnapping Stiles, he can get some sort of revenge or leverage.
Well, it could be an envious maid from the palace. She sees Cinderella getting all the love and attention from the prince and decides to kidnap her. Maybe she thinks that if Cinderella is out of the picture, she can win the prince's heart. Or perhaps it's a dark wizard who has been observing Cinderella from afar. He wants to use her pure heart for some dark magic ritual, so he kidnaps her. There's also a chance that it could be a group of outlaws who heard about Cinderella's new wealth and status and think they can get a big ransom for her.
It could be Frieza. He might see Bulma as a valuable asset due to her intelligence and knowledge of Earth's technology. He could kidnap her to use her to build powerful weapons or devices to help him regain his power and dominance in the universe.