Every state does not have online schools, but Florida does. I think you would need to tell a little white lie, and declare Florida as your primary residence, but once you are considered living in Florida you can enroll your children in a virtual/online public school, fully paid for by public funds. Sort of like homeschooling, but still a "real" public school in that you get a HS diploma when you are done. Sounds like an ideal situation for someone that wants to be in Florida 1/2 the year.
We chose full-time homeschooling (for a lot of reasons), but high among them was the ability to go where we want, when we want, and not have to deal with the school system at all. Its not an easy choice, but it gives you ultimate flexibility.