Another possibility is that it's a group of thieves. They might have heard about Skye's special abilities and think that they can use her to steal valuable things in Adventure Bay. For example, they could make Skye fly them over to the highest security vaults or something. So they kidnap her to force her to do their bidding.
It could be Mayor Humdinger. He's always causing trouble for the Paw Patrol and might see kidnapping Chase as a way to get back at them.
Well, in the fanfiction, the other Paw Patrol members would surely start a search right away. Chase might use his police skills to look for clues. They could find Skye's pup tag or some fur at the kidnapping site. Then they'd follow the trail to rescue her.
Probably the other pups in the Paw Patrol would come to Skye's rescue. Rocky might use his recycling skills to build something useful to help in the search, like a makeshift ladder if Skye is stuck in a high place. Zuma could swim across any water bodies in the area where Skye is lost to reach her. And Rubble might dig through obstacles if needed.
Well, in this fanfiction, Skye getting lost could lead to a big adventure. The other Paw Patrol members would surely start a search for her. Skye herself might try to use her flying skills to find her way back. She could also find some hidden treasures or secrets in the place where she is lost, which could be exciting for the whole Paw Patrol when she's found and shares her discovery.
One possible plot could be that Skye is on a mission to rescue a stranded hiker in the mountains. She misjudges a strong wind current and crashes into a rocky outcrop. The other Paw Patrol members then have to come up with a plan to rescue her and treat her injuries. They might face challenges like bad weather and difficult terrain while trying to reach her.
Skye getting sick could also lead to a new adventure. Maybe the cause of her illness is related to some mysterious thing in Adventure Bay. The pups would then have to work together to find out what it is and how to stop it from making Skye or any other pup sick in the future. They might search all over the bay, from the lighthouse to the beach, following clues and using their unique skills.
Perhaps it's an organization that wants to use Kakashi for their own evil plans. They might think that Kakashi's skills and knowledge would be useful for them. For example, they could be trying to create some sort of super - weapon and they need Kakashi's expertise in ninjutsu.
It could be Hades. He's always been an antagonist in the series and might kidnap Pit to mess with the gods. He could be thinking that by taking Pit, he can gain an advantage in his ongoing feud with them.
It could be a rogue Saiyan who wants to take over the Saiyan throne and thinks kidnapping Goku is the first step.
It could be a new villain that we haven't seen in the main Mario games. Maybe a powerful sorcerer who needs Mario for some dark magic ritual. This sorcerer might have been lurking in the shadows of the Mushroom Kingdom, waiting for the right moment to strike and kidnap Mario.