Maybe it's a cult leader. The cult might believe that Reid has some sort of special knowledge or power that they want to harness. They kidnap him to convert him to their beliefs or use his mind for their own strange purposes.
It could be a copycat unsub who idolizes the previous criminals the BAU has caught. He might have studied Reid's work and decided to target him to make a name for himself in the criminal world.
The kidnapper could be a power - hungry villain from another dimension. This villain has heard of Dr Flug's reputation and thinks that by kidnapping him, he can gain access to advanced technology. Or it could be a group of bounty hunters who were hired by someone who has a grudge against Dr Flug. They would capture him and deliver him to the person who hired them.
Well, perhaps in the fanfiction, Reid is kidnapped as part of a plan to blackmail the BAU. The kidnappers demand that the team stop investigating a certain case or they'll harm Reid. It could also be that Reid is kidnapped by a group who believe he has information about a secret government project. Reid then has to figure out how to convince them he doesn't while also finding a way to get free.
It could be someone Reid put away in the past. Maybe a criminal who has held a grudge against him ever since and has now found an opportunity to kidnap him for revenge.
It could be a deranged fan of one of Reid's academic works. This person might have misinterpreted Reid's theories and decided that Reid was some sort of false prophet. So, in his twisted mind, he kidnaps Reid to 'correct' him.
One possible plot twist could be that the kidnapper is someone from Reid's past that he thought was long gone. Maybe an old unsub who escaped prison and is seeking revenge. Another could be that Reid is kidnapped as a means to get to the BAU team, and the kidnapper is using him as bait to draw out the whole team for a bigger attack.
One possible motive could be revenge. Maybe Reid has put away a relative or someone close to the kidnapper in the past. Another motive might be related to Reid's intelligence. The kidnapper could want to use his brainpower for some illegal or self - serving scheme, like cracking a code or coming up with a complex plan.
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.