It could be Chiron, the centaur. He's always keeping an eye on the campers and would likely be concerned if he caught them up to something. He might give them a stern talking - to and some extra chores as punishment.
It could be their friends like Grover. Grover is always around them and he might accidentally stumble upon them while they are in a private moment.
Perhaps they were caught in a compromising position while having a private moment. This could lead to all sorts of awkwardness among their friends at the camp. They might have to face some teasing or even get in a bit of trouble with the camp counselors for not following the rules.
It could be that Annabeth has a misunderstanding about Percy. She might think he's a threat to her position or status at the camp, so she bullies him to try and keep him in check.
Yes, there are likely many fanfictions out there with this scenario. Fanfiction websites like Archive of Our Own (AO3) and FanFiction.net probably have numerous stories about Percy and Annabeth where they are caught kissing. You can search using their names and relevant keywords on these platforms to find such fanfictions.
One challenge could be office gossip. People might start talking if they notice Percy and Annabeth getting too close. This could create a difficult work environment for both of them. Another challenge is the power dynamic. Percy has to be careful not to abuse his power as a boss when it comes to his relationship with Annabeth.
Annabeth could try to be more understanding of Percy's needs. In the fanfiction, she might give him space at first, but then slowly show up in his life in small, positive ways. Like bringing him his favorite food or helping him with a difficult task without being pushy.
They might cuddle because they've been through a lot together. All the battles and quests have made them rely on each other. At camp, it's a place of relative safety where they can finally let their guard down and be affectionate.
Grover could be one of Percy's friends to meet Annabeth. He's a satyr and has been with Percy through a lot. When he meets Annabeth, he might be really happy to see another important person in Percy's life.
It could be a new demigod who has a grudge against Percy and somehow convinces Annabeth that Percy is a threat, leading to her tying him up. So this new demigod would be the antagonist.
There could be many authors. It might be a new writer in the fanfiction community who wants to explore this rather dark and emotional side of their relationship.