Well, one possible plot could be that the Akatsuki kidnap Hinata to use her Byakugan for their own purposes. Maybe they want to study it to gain an edge in their battles or espionage. They might hold her captive in one of their secret hideouts, which could be an old, abandoned fortress.
There's also a possibility that Hinata's own clan, the Hyuga, would take matters into their own hands. They have their own unique techniques and a sense of pride. They would not allow the Akatsuki to keep one of their own captive for long and would send their strongest members to retrieve Hinata.
In a 'hinata shouyou kidnapped' fanfiction, the plot might involve Hinata being kidnapped by a mysterious figure who has a vendetta against his volleyball skills. This person wants to break his spirit. Hinata is initially scared but then realizes he can't give up. He starts to look for weaknesses in the kidnapper's hideout. Meanwhile, his coach and team notice his absence and suspect foul play. They follow some clues that Hinata cleverly leaves behind and are able to rescue him. After his rescue, Hinata comes back stronger than ever, determined to show the kidnapper that he can't be broken.
Perhaps Hinata is kidnapped because she has some unique power or knowledge that the kidnappers want. In the fanfiction, the story could focus on how she uses her wits and hidden strength to try to escape while her comrades search for her. For example, she might be able to communicate with a small animal in the cell she's held in and use it to send messages.
In a 'Sakura and Hinata joins Akatsuki' fanfiction, perhaps they are forced to join. The Akatsuki could be holding something or someone important hostage. Sakura might be trying to find a way to break free from the inside, using her intelligence and combat skills. Hinata, on the other hand, would be more conflicted due to her gentle nature. She might befriend some of the more empathetic Akatsuki members and try to persuade them to change their ways.
Well, in such a fanfiction, Sakura might be used as a bargaining chip. The Akatsuki could try to exchange her for something valuable, like secret jutsu. Or they might want to extract information from her about Konoha's defenses.
The plot might involve Sakura and Hinata being deceived into joining the Akatsuki. They initially believe they are joining a group that has good intentions. Once they realize the true nature of the Akatsuki, they have to decide whether to stay and try to change things or escape. Sakura's medical skills could be useful to the Akatsuki in treating their members, and Hinata's gentle nature might start to influence some of the more brash Akatsuki members.
In some fanfictions, Naruto could be a double agent in the Akatsuki. His goal might be to infiltrate and destroy the organization from within. Hinata could be his support, using her Byakugan to gather intelligence. For example, she might be able to detect hidden Akatsuki bases or members who are trying to ambush Naruto.
The plot could involve a lot of internal conflict within the group. Sakura, Ino, Hinata, and Tenten would bring their own values which are quite different from the typical Akatsuki members. For example, Sakura's strong belief in medical ninjutsu for healing rather than destruction. This could lead to power struggles within Akatsuki as they try to fit in or change the group's ways. Also, their relationships with the male characters in their original villages would be put to the test. Naruto, for instance, might be shocked and try to convince Sakura to leave Akatsuki, creating a complex web of emotions and storylines.
One possible plot twist could be that Sasuke was actually being controlled by an unknown enemy when he kidnapped Hinata. This enemy might be using Sasuke's power to get to Naruto through Hinata. Another twist could be that Hinata has some hidden power or knowledge that Sasuke's side needs, and the kidnapping is a way to extract that from her. Maybe it turns out that Hinata was expecting the kidnapping all along as part of a larger plan to infiltrate Sasuke's group and learn their true intentions.
If Sakura gets kidnapped by the Akatsuki in fanfiction, Sasuke could also get involved. Since he has his own history with the Akatsuki, he might use his knowledge to track them down. Also, Sakura herself could be trying to find a way to escape from within. She's a strong kunoichi, after all. She might be gathering information about the Akatsuki's plans while being held captive.