(A) which caused only a mild illness, they are immune from
(B) causing only a mild illness, they become immune from
(C) which causes only a mild illness, they are immune to
(D) causing only a mild illness, they became immune from
(E) which caused only a mild illness, they would become immune to

Splits
B&D out - Usage of verb-ing modifier incorrectly modifies the entire preceding clause. We need to modify only the preceeding noun entity i.e. cowpox. Correct usage of verb-ing modifier here would been sans the comma.
A out - Incorrect usage of verb-tense. The observation was in the past, the effect was with a time lag. Happened -- Would become
C out - Idiom problem. Immune from is incorrect.

E is the winner.