A rival within the police force or someone who wants to discredit the Reagans could also be the kidnapper. They might think that by kidnapping Jamie, they can make the family look bad and gain an advantage in some internal police politics. Maybe they are jealous of the Reagans' reputation and want to bring them down a notch.
There's also a chance that it's a complete stranger who has a vendetta against law enforcement in general. Jamie, being a part of the Reagan family which represents the police force, becomes an easy target. This person might have had a bad experience with the police and decides to take it out on Jamie as a symbol of all that they hate about law enforcement.
One possibility is that the family will mobilize all their resources to find him. In the Blue Bloods world, they have connections in the police force and other areas. They might start by retracing Jamie's last known steps. Maybe he was lured away by someone with a vendetta against the family. The kidnapper could be using Jamie as a pawn to get to the Reagans, perhaps for some past case they were involved in.
Regarding a fanfic on Jamie's shot in Blue Bloods, it might focus on the emotional impact on Jamie himself. He could be haunted by the moment of the shot, constantly replaying it in his mind. In the fanfic, he might seek advice from an older, more experienced colleague who has been in a similar situation. This colleague could share their own story of when they took a shot and how they coped afterwards. It could be a story of growth and learning for Jamie as he tries to come to terms with what he has done.
Revenge is another motive. Maybe someone has a grudge against the Reagan family for past actions. Nicky is an easy target to get back at them. For example, if the family put someone in jail, that person's accomplice might kidnap Nicky to make them suffer.
If Eddie was kidnapped in a Blue Bloods fanfiction, she would surely try to escape on her own. Eddie is a tough character. She might use her training to overpower the kidnappers or at least cause enough of a distraction to get away. Maybe she'd find a weapon in the place where she's being held and use it to her advantage.
One idea could be that in the fanfiction, Jamie's family rallies around him after he gets shot. They use their combined skills and influence within the police force to track down the shooter quickly. Each family member has their own moment of realization about how much Jamie means to them.
It could be someone from a case he was working on. Maybe a criminal he put away in the past who has just been released and is seeking revenge. They might have hired some thugs to carry out the beating to send a message to Jamie and the police force in general.
It might be an escaped criminal who just happened to come across the boy while he was being tickled. The criminal, seeing an opportunity, kidnaps the boy. The boy's urinating could be a sign of his helplessness and the terrifying situation he has been thrust into by this unexpected kidnapper.
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.