While the General himself couldn't make the nomination, all he needed to do was call a friendly congressman who had a slot available. The West Point wiki article says a nominee doesn't need to know the congressman. I read that to mean a potential nominees doesn't need to reside in the congress critters' district.
When General Bill Westmoreland says he can get you an appointment he can get you an appointment. General Westmoreland was Army Chief of Staff (the highest ranking General in the Army) from 1968 to 1972. The conversation too place the summer before Carson entered Yale in 1969.
At the time Westmoreland told Carson he could get him an appointment to West Point Westmoreland was the highest ranking General in the US Army, and was the Commandant of West Point's boss. When the Chief of Staff says he can get you an appointment to West Point you can take it to the bank. It was a legitimate offer. Westmoreland did not blow smoke.