A car should normally be registered in the US state in which it is garaged (aka parked when not in use). If this car is to be kept and used in West Virginia for an extended period (several months or more), you should have it registered there. This is legal. In fact failing to do so might be a violation of the law.
It is legal to allow any licensed driver to use your car if you choose to. That certainly includes your wife (provided that she is properly licensed).
You should notify the insurance company where you have a policy on the car of the changed address. You should make sure that your policy complies with at least the minimum requirements in West Virginia, which may be different from those in Texas. You should also notify the insurance company that your wife will be driving the car regularly, with an estimate of the average daily or weekly mileage that the car will be driven. All this may change the premium on your policy.
The state's DMV site should provide a list of requirements for a car to be registered there. Follow it. An inspection may be required.
