Having the right hand and arm being crippled by a sniper’s bullet during the First World War, Horace Pippin, a Black American painter, worked by holding the brush in his right hand and guiding its movements with his left.
(A) Having the right hand and arm
being crippled by a sniper’s bullet during the First World War
Wrong. "being crippled" is incorrect, because "crippled by a sniper's bullet" modifies "hand and arm" ==> Verb+ed is correct, not "being Verb+ed".
I don't think "having" is a problem here. For example: having a car towed yesterday, Harry comes to school late today.
(B) In spite of his right hand and arm
being crippled by a sniper’s bullet during the First World War
Wrong. Same as A. "being crippled" is incorrect modifier.
(C) Because there had been a sniper’s bullet during the First World War
that crippled his right hand and arm
Wrong. "that" seems to modify "first world war". In addition, the structure is very awkward "there had been.....that crippled.....".
(D) The right hand and arm
being crippled by a sniper’s bullet during the First World War
Wrong. Same as A. "being crippled" is incorrect modifier.
(E) His right hand and arm crippled by a sniper’s bullet during the First World War
Correct. "his" refers to Horace Pippin, "crippled" modifies "hand and arm".
Hope it helps.