If your family story is correct, that would indicate that he had been captured. Vicksburg was in Union hands at the end of the war (and had been since the Fall of Vicksburg in July 1863). It was a major prisoner exchange site. There were no Confederate units there, except for those across the river facilitating the exchanges.

BTW, the 36th Ala. was not at Vicksburg when it fell in 1863, so your family's story would not be confusing that event with the end of the war. I say that because a lot of family stories do confuse that event with the end of the war.