Let's evaluate each answer choice:
Original sentence:
Based on accounts of his contemporaries, academicians have reconstructed a rough timeline of the life and career of William Shakespeare, who left behind virtually no firsthand accounts of his childhood in Stratford-upon-Avon or career in London.
(A) Based on accounts of his contemporaries
The original sentence seems to be grammatically correct, and it concisely and effectively conveys the idea that scholars have been able to piece together a rough timeline of Shakespeare's life and work primarily from the accounts of his contemporaries, since Shakespeare himself left behind very little direct evidence of his life and career.
(Let's Keep this for now)
(B) Basing it on his contemporaries’ accounts
The pronoun "it" has no clear antecedent. This sounds a bit awkward. The word 'basing' is not used correctly.
(Eliminate)
(C) With accounts of his contemporaries used for a basis
This choice is awkwardly phrased and a bit redundant. "with accounts" sounds a bit less direct.
(Eliminate)
(D) Using accounts of his contemporaries
I think, this choice is also grammatically correct and conveys a clear meaning. It uses a different preposition ("using" instead of "based on") but the meaning is essentially the same.
(Let's Keep)
(E) By the account of his contemporaries they used
This sounds weird and un-grammatical. The phrase "they used" has no clear antecedent and makes no sense in context.
(Eliminate)
Between A and D, both sentences are grammatically correct and effectively convey the same meaning and it might be tough to eliminate one. However, I think (A) might be preferred because, "Based on accounts of his contemporaries" sounds slightly more formal and academic in tone than "using accounts of his contemporaries," .
So, I will go with (A).
Waiting for OA.