My family has just purchased a house. It was built in 1849 and has never had a washing machine. There is a mud room next to the bathroom with a crawl space under both. We would like to install a washer hook up in the mud room by ourselves using the water from the bathroom. How would we go about doing this? For instance- materials, permits( if any), and over view on what needs to be done?

Permits is up to your city building inspector so call him. Sounds lkike you have your water supply close by but do you also have a drain line at least 2" in diameter? What kind of supply pipes are in the bathroom, copper, galvanized, PVC?