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 possible plot could be that they are in a restricted area of Camp Half - Blood. Maybe they thought they were alone, but a younger camper stumbles upon them. This causes a bit of a stir among the campers, with some being happy for them and others thinking they were being inappropriate. Percy and Annabeth then have to deal with the fallout, like facing the camp counselors.
In the fanfiction world, if Percy and Annabeth were caught kissing, it could lead to various reactions. Their friends might tease them mercilessly. For example, Grover could make some funny goat - like noises and give them knowing looks. It could also create some tension within the group if there are those who secretly have feelings for either Percy or Annabeth. Maybe some would be jealous or feel left out. Another possibility is that their relationship becomes more public knowledge, which could have both positive and negative impacts. On one hand, they might get more support from their friends who are happy for them. On the other hand, it could attract the attention of enemies who might try to use their relationship against them.
Percy would most likely go into a state of panic. His mind would be racing, thinking about all the teasing he'll get from his friends and what his mother would say if she found out. He might try to make a joke to ease the tension, but it would probably come out all wrong. Annabeth would be a bit more calm, but still mortified. She would be thinking about how this could affect their reputation in the camp, especially since they are both leaders in a way. She might give Percy a look that says 'this is all your fault' even if it wasn't really.
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.
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 signify deep affection. In fanfiction, such a gesture might show that Percy is very comfortable and close to Annabeth, and it's a more intimate form of showing his love compared to a simple cheek or lip kiss.
Perhaps in a moment of peace at Camp Half - Blood. They might be walking by the lake, enjoying the tranquility, and suddenly, overcome with their feelings for each other, they kiss. It shows their relationship growing in a more calm and domestic setting within the camp.
Well, one consequence could be that other students might see them and start gossiping. This could either make them the talk of the school in a positive way, like a cute couple everyone adores, or in a negative way if there are some who don't like them together. Another possibility is that their friends might react differently. Some friends might be really happy for them, while others might be a bit shocked or jealous. And it could also affect their own relationship. Maybe they become closer because of this public display of affection, or maybe they start having some insecurities about how fast things are moving.