Oberlin College in Ohio was a renegade institution at its 1833 founding for deciding to accept both men and women as students.
Option Elimination -
(A) at its 1833 founding for deciding to accept - "founding in 1833" is better than "1833 founding" as it can mislead us to think as if they had multiple founding. Moreover, "for" is used to imply the purpose of an action. The sentence, as written, "founding" is a gerund, and the prepositional phrase "for deciding to accept..." modifies the gerund. Here, we aren't trying to say that the purpose of "founding" was to accept both men and women. The intended meaning is that the "renegade" nature of the institution stemmed from the decision. So, we need a prepositional phrase to modify the verb phrase "was a renegade institution." A better option would be "Oberlin College was a renegade institution in deciding to accept both men and women at its 1833 founding" or even better, "Oberlin College was a renegade institution in deciding to accept both men and women at its founding in 1833."
(B) for the decision at its 1833 founding to accept - "for the decision" is an issue here. It conveys the meaning that "Oberlin College in Ohio was a renegade institution" because of "the decision" itself. While the intended meaning is "Oberlin College in Ohio was a renegade institution" because of "an act of making a decision." So we could have said it like this to sound better: Oberlin College was a renegade institution for deciding at its 1833 founding to accept both men and women as students, or even better, "Oberlin College was a renegade institution for deciding at its founding in 1833 to accept both men and women as students."
(C) when it was founded in 1833 for its decision to accept - Wrong sequence of events. It conveys "it was founded," and then maybe the decision was taken. The intended meaning is that the decision was taken at the founding.
(D) in deciding at its founding in 1833 to accept - ok
(E) by deciding at its founding in 1833 on the acceptance of - "by deciding" is inferior. The act of making the decision is the cause of the renegade nature rather than being the means or method by which Oberlin College achieved its renegade status. Moreover, "on the acceptance" conveys as if the decision was focussed on the act of acceptance while the intended meaning is that the decision was to accept men and women and not focussed on the acceptance itself. "to accept" is better.