Nope. Politico is lying, and you are falling for it!
Here is Carson's account. Kindly point out the lie.
At the end of my twelfth grade I marched at the head of the Memorial Day parade. I felt so proud, my chest bursting with ribbons and braids of every kind. To make it more wonderful, We had important visitors that day. Two soldiers who had won the Congressional Medal of Honor in Viet Nam were present. More exciting to me, General William Westmoreland (very prominent in the Viet Nam war) attended with an impressive entourage. Afterward, Sgt. Hunt introduced me to General Westmoreland, and I had dinner with him and the Congressional Medal winners. Later I was offered a full scholarship to West Point. I didn’t refuse the scholarship outright, but I let them know that a military career wasn’t where I saw myself going.
It's no mystery why West Point has no record of Carson applying for admission. He applied to exactly one college, Yale, and was accepted.
BTW, some morons are picking nits about the use of the word scholarship. In point of fact, West Point itself calls what it offers a scholarship — a full government scholarship, no less:
Of course, admittedly, being the gubmint, they are not required to make full disclosure in their advertising. Hence the omission of the post-graduation service commitment.