As the Office of the Insurance Commissioner for Washington state notes, the state requires anyone that operates a motor vehicle to have liability insurance. There is no grace period for liability.
The insurance regulator for the state of Washington goes on to say that all insurance companies have different time requirements to add a newly acquired vehicle to the policy. You would need to check with your insurance carrier to find out if there is a grace period for adding that new vehicle to your existing policy. Most dealers require physical damage coverage on the vehicle before it leaves the dealership.
So if you already have insurance on other vehicles and purchase a new vehicle that needs to be added to your insurance policy you should contact your insurance agent to see if your policy will extend to the new car for a certain amount of time or if you must add the car immediately to the policy to be covered. Many times if you are replacing a vehicle on your policy with the new vehicle then the coverages of the old vehicle would extend to your new vehicle for at least a few days.
If you do not already have an auto insurance policy on any vehicles but are buying a car then you will need to obtain insurance before leaving the lot with the car normally. You can give an insurance company information on the vehicle ahead of time so that the policy can be started. The dealership may need to fax or contact the insurance agent with information as well. If for some reason you do not end up buying the car then you would inform the insurance company so that the policy would not go into effect.