RVer4Life, you are looking at a wonderful opportunity for your family. This will take some thought and prayer as you find the best State for your new domicile. The easiest way to handle things is to cut all ties with your current State. As far as your current school district is concerned, all you need do is tell them that you are moving to a new State (whichever one you pick). They don't need to know, nor do they care, what you will be doing there.
Probably the easiest way to avoid running afoul of any school regulations is to not stay in one place for more than a couple of weeks. If you move to a new location every 7-10 days there should be no problem. You should research the laws in every State you visit before you set foot in it.
One caveat: NEVER tell anyone you are working, as that will trigger residency requirements. Those include, but are not limited to, vehicle registration, drivers licenses, home-school laws, etc. You are visiting the State.
Enjoy the experience. We homeschooled our three children and wished we could have done so while seeing the country.