Are you required by law to let the insurance company take your car if it is totalled before their appraiser has appraised it?

No. I would suggest that you insist that they don't move your car. They will approach you to ask if they can move the car to a lot that doesn't charge storage. If you do that you have given away some of your leverage. You do not have to let them move the totalled vehicle until they have made you an offer. The day after an offer was made to you, they are no longer responsible for the storage charges, you are. You need to look around and see what similar vehicles are selling for at car dealerships and in personal ads so that you know the ballpark your vehicle is worth. Most adjusters want to finish the claim, pay you a fair value and get the claim closed with everyone happy. If you have reasons your vehicle is worth more than average tell the adjuster. If you have just put on a new set of tires, have very low mileage, etc. these can increase your value. If your is very clean from scratches or never being hit before, etc.