Each state will have it's own rules and regulations that it uses to determine whether or not you are a resident for income tax purposes. There is no uniformity. If a fulltime RVer is going to work in a state, or owns real property in a state, has registered vehicles in a state, or will spend a significant amount of time in a state, it would be advisable to be aware of the tax laws considering residency in that state. Most states have a publication that explains their residency rules. Many states will have this information available on a web site.
All taxpayers are responsible for keeping tax records.
It may just be a little more challenging for a fulltime RVer to find the space. RVers who have a possibility of state taxation as nonresidents or part year residents should keep especially good records detailing time spent in each state. The Internal Revenue Service requires you to keep your tax records for a minimum of three years after the due date of the original return. Many states add an additional year to that time period. I would recommend keeping the actual tax return copies, W-2's, and cancelled checks for payment of tax for seven years.