As rare as something becomes, be it a baseball card or a musical recording or a postage stamp, the more avidly it is sought by collectors.
(A) As rare as something becomes, be it
(B) As rare as something becomes, whether it is
(C) As something becomes rarer and rarer, like
(D) The rarer something becomes, like
(E) The rarer something becomes, whether it is
"like" should not be used to enumerate. Eliminate C and D.
"whether it is" is correct. Eliminate A.
"the rarer something becomes" is correct as it shows movement of becoming more rare "rarer" which is the intended meaning of the sentence: the rarer, the more avidly it is sought...
Impossible is nothing to God.