How can a person have a clogged basilar artery when her carotids are clear?

Just is. The carotid arteries are anterior circulation vessels; you can think of them as supplying the front of the brain. The basilar artery is at the base of the brain and is created by the confluence of the two vertebral arteries, which are posterior circulation vessels. It is possible to have disease in only one of these vessels, or in all of them, or in any combination.
Clot moved to there. The two most likely is congenital blockage you had entire life or a clot (usually fro a-fib) mover to block it.