It can be done. I've been involved on jobs where we jack up and support one wall at a time with timbers and screw jacks, digging out the basement wall with a backhoe, pour the footing, and then pour the wall. Work your way around the house. Before the last wall, dig out from under the house, being carefull to pour footings and jack up and support the middle of the house. Most usually pour the floor last through the basement windows. It is like replacing the foundation but just a taller foundation. It is hard work but doable.
Some will jack up and support the entire house with beams as if they were moving the house. Then dig the basement all at once. I even know one contractor who will move the house out of the way if he can. You will need a basement contractor unless you have some experience at it along with the equipment and basement forms. Good luck.