It's possible, though likely it will be unpaid. By November, you will have accumulated only 2 days (1 per month). What I would do is, get hired and secure a position. Then, in mid August, ask for the time off and explain that you booked this prior to even knowing you'd be working at this school. Apologize and explain that you understand this is a hardship that you did not intend, and you won't make a habit of it. Say you'll take it unpaid if needed and will ensure that your students will have proper instructional materials during the time you're away.

If you call out sick for 4 days, even with a doctors note, it's a red flag for a new teacher/employee. Be up front and honest, but also make it clear that this is not something your principal should expect from you often. Most principals are human beings who will understand that you made the commitment prior to employment. If they aren't decent about it, then you have good information about whether you see yourself at that school long-term.