Unable to walk without assistance, radio provided Franklin Roosevelt with a medium for conveying a message of vigor and confidence that would have been belied by the visual image presented on television.

(A) radio provided Franklin Roosevelt with a medium for conveying
Woking since modifier "unable to walk ..." is illogically modifying radio.
(B) Franklin Roosevelt conveyed in radio
Here in is not a correct preposition. Also there is a meaning change as importance of radio is lost here.
(C) Franklin Roosevelt used radio to convey
Correct
(D) radio allowed Franklin Roosevelt to convey
Same error as in option 1
(E) through radio it was possible
Same error as in option 1

KAPLAN OFFICIAL EXPLANATION:

C

You don’t want to say that radio was unable to walk without assistance.You want to say that Franklin Roosevelt was. So the noun “Franklin Roosevelt” must follow the intro phrase, as it does in (B) and (C). In (B), it’s wrong and awkward to say that Roosevelt “conveyed in radio a message.” (C) puts it best: Roosevelt “used radio to convey a message.”
_________________