There's no waiting period, and paperwork on a car shouldn't be an issue. Newcomers can buy insurance in their new market as soon as they move; all that's needed is a permanent home address (read more here).
In fact, Barbee didn't need to wait until 2015 open enrollment started last month. Any time you move into a new market -- and that can mean simply crossing county lines -- you're eligible to apply for a subsidy and buy insurance on the Affordable Care Act exchange. Once you've bought insurance, be sure to go back to HealthCare.gov and enter your new address if you move; it may change your options.
Meanwhile, the clock is ticking for 2015. Dec. 15 is the deadline to select coverage that takes effect on Jan. 1. That includes not just first-time buyers but people who want to make changes to their 2014 policies. I've reported before that it's smart for everyone who already has a policy to check out the new options; read more about the hazards of health-insurance inertia from the New Yorker's James Surowiecki.