Early medieval monasteries, while clearly less accessible to outsiders, often served as repositories for texts like modern libraries.
(A) while clearly less accessible to outsiders, often served as repositories for texts like modern libraries
(B)
like modern libraries, often served as text repositories, though they were clearly less accessible to outsiders
(C) while clearly less accessible to outsiders, often served as repositories for texts as does
the modern library(D)
like modern libraries,
while clearly less accessible to outsiders, often served as repositories for texts
(E) while clearly less accessible to outsiders, acted like modern libraries act by serving as repositories for texts
(B) & (D) are out right away because "while clearly less accessible" is misplaced.
Monasteries are less accessible; modern libraries are not less accessible, in fact, they should be more accessible.
This is the intended meaning of the sentence. Monasteries have similar functions modern libraries do, but monasteries were less accessible.
(C) is out because at the end of the sentence it changes to a singular library. It's comparing monasteries (a general term) to "the modern library" (a specific).
Between (A) and (E), (A) just seems better.