The kidnapper could have some sort of motive. It could be for ransom, in which case her family would be contacted. Or it could be something more sinister, like the kidnapper has a personal vendetta against her. The story could then focus on how Lucy tries to escape from the kidnapper's clutches, using her wits and any resources she can find in the place where she's being held.
A detective character could also be involved. They are trained to solve mysteries and find missing people. They would use their investigative skills, interview witnesses, and follow any leads. Maybe they've been on similar cases before and have some experience in dealing with kidnappers.
In general, it's not acceptable. Such themes are not only disturbing but can also have a negative impact on the perception of the original work. Fanfiction should be a place for creativity that is within the bounds of good taste and respect for the source material. Torture and kidnapping are serious and dark topics that are better left out of fan - created works.
One potential consequence is getting a negative reputation within the fanfiction community. Readers may avoid the author's other works. Also, it could be flagged as inappropriate content on many platforms.
A new, evil sorcerer who has just arrived in Narnia could be the kidnapper. He might have heard tales of Lucy's importance and figure that by taking her hostage, he can control the forces of good in Narnia. Or perhaps a tribe of wild centaurs who have been corrupted by some dark magic might kidnap Lucy. They could be under the influence of a spell that makes them believe Lucy is the cause of all their problems and that getting rid of her will solve everything.
In a fanfiction where Lucy is kidnapped, the Fairy Tail guild would likely spring into action. Natsu, being her closest friend, would be furious and determined to find her. He'd probably charge off without much of a plan, with Happy flying after him trying to keep him in check. Gray would also be involved, using his ice - magic to combat any obstacles in their way. Erza would lead the strategic part of the rescue mission, planning their every move carefully.
In some fanfictions, if Natsu kidnapped Lucy, it could be a comical situation. Natsu might do it out of his own strange sense of protecting Lucy in an overzealous way. Maybe he thinks there's a danger that only he can keep her safe from, and his impulsive nature leads him to this 'kidnapping' which is more like a forced protection.
One way it could end is with a big showdown. The Fairy Tail guild members find the kidnappers' lair, and there's an epic battle. Natsu defeats the main kidnapper with his powerful fire magic, and Lucy is freed. Then they all return to the guild, where there's a big celebration.
One potential kidnapper could be a dark wizard who wants to use Lucy's Celestial Spirit keys for their own dark magic. Another could be a rival guild member who is jealous of Lucy's power and wants to steal her keys to gain an advantage in the magical world. It could also be someone from Lucy's past, like an old acquaintance who has turned bad and has some grudge against her.
One idea could be that Lucy is kidnapped by an old enemy of Narnia who wants to use her as a bargaining chip to get something from Aslan. For example, they might demand that Aslan leave Narnia forever. Another plot could involve a group of dark creatures kidnapping Lucy because they believe she has a special power that they can harness for their own evil purposes. Maybe they think her connection to Aslan gives her some sort of magical energy that they can steal.