A site once used as an observatory by the Anasazi, ancient pueblo dwellers of New Mexico, has been recently discovered where patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle.(A) A site once used as an observatory by the Anasazi, ancient pueblo dwellers of New Mexico, has been recently discovered where patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle. -> Is ancient pueblo a site once used....Incorrect.
(B) A recently discovered site was once used as an observatory by the Anasazi, ancient pueblo dwellers of New Mexico, where patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle. -> Same as A. Incorrect.
(C) At a recently discovered site once used as an observatory by the Anasazi, ancient pueblo dwellers of New Mexico, patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle. -> Now the first part is changed, it starts with a preposition and meaning wise sentence is better too. Let's keep it.
(D) Patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle at a site that was recently discovered and was once used by the Anasazi, ancient pueblo dwellers of New Mexico. -> "cycle at a site" is incorrect. It changed the intended meaning.
(E) Patterns of light and shadow were employed to establish the precise limits of the positions of the Sun and Moon over a nineteen-year cycle at a recently discovered place that the Anasazi, ancient pueblo dwellers of New Mexico, once used the site as an observatory. -> Modifier error, "discovered place that the Anasazi, ancient pueblo dwellers of New Mexico".
So, I think C.