No. You cannot recieve an appointment without a nomination. From what I understand a nomination gives USMA the legal authority to accept you into their academy. Even if you get a LOA, you still have to get a nomination and be medically cleared in order to finally recieve an appointment.
Also, beginning last year they really stopped giving out a lot of LOA's, and I expect by the time you apply they will give out almost none. Instead they give out Letters of Encouragement (LOE), which do not carry the same level of commitment from West Point as the old LOA did. If you get one it means USMA is impressed with you, but it does not mean they will reserve a seat for you as they would for someone with a LOA.