Yes,
You can do this. Member Services will reserve a car for you even without a credit card. Alamo/National is at the Airport but they also have a location right by the Magic Kingdom. You can rent a car for any of the days you need and they will pick you up at your hotel and drive you to the rental location. Or, if you are taking a flight in, you can have Member Services check for a car at one of the airport locations. They have all the rates of the local companies and can give you rates for the dates you are interested in. I have reserved through Member Services a few times. One thing I always do is check the MS rate against what I can find on the Internet. So far, I have not found rates that differ than what MS provides me.