When you drive in Washington State, you are required to have insurance. If you drive a vehicle that is registered in WA you must have an auto insurance policy with limits of at least: $25,000 of bodily injury for one person in any one accident and $50,000 of bodily injury of any 2 persons in any one accident plus $10,000 of injury or destruction of property of others in one accident (property damage liability).
Typically, you will either need an drivers license or be able to get one within 30 days in order to get insurance on a vehicle. Also, with only have a learner's permit, the insurance company likely will require your daughter to take out the policy with a driver on it who has a full, valid driver's license.
Due to the restrictions most insurance companies have, if they even offer a policy to a permit holder, you may instead want to add your daughter and the car to your current insurance policy. Your insurance company should be informed that she has a learner's permit at this time anyway since most insurance providers require that any household members with any type of licensing status be reported to them.
If you would rather your daughter has her own policy on the car you gave her then after she obtains her license this should be easier for her to do. Keep in mind that if she is under 18 normally insurance companies will require a parent or guardian to sign on the insurance policy with her since she is a minor. If this is the case then you want to consider keeping her and her car on your policy which may be more economical.
For a Washington State insurance quote, click here.