The reason that you are getting short vacations in the summer is because Korean summer break is much shorter than their winter break, so to plan around camps they want you to take short vacations in the summer.
You can ask if you can have more, but if the camp is already set then they won't want to change it.
You might suggest doing two camps in one day, which makes it a busy day, but you could get more vacations (and maybe overtime pay)
Our school is having us teach one camp in the morning and then another in the afternoon, because we wanted to take as few vacations as possible in the winter because we had family coming in town in the summer.
Your issue is that you haven't had winter break yet so you don't have any entitlement to vacations, so you might just have to deal with it this time around.
Read your contract, I think mine says that winter and summer vacations are more or less split evenily, if that's the case, maybe you can argue for 2 weeks.
Good luck!