(A)
a child born into a family whose members have allergies will probably
themselves develop allergies
=> mismatch as "a child" is a singular noun while "themselves" is a plural reflexive pronoun
(B) children born into families whose members have allergies will probably themselves develop allergies
=> correct use of relative pronoun "whose" and reflexive pronoun "themselves" which refers to "children" (C) a child born into a family the members
of which have allergies will probably develop an allergy
=> the use of "of which" is not approriate => a "whose" between "family" and "members" is preferred
(D) in those families
where members have allergies, children will probably develop allergies themselves
=> "those" is quite a redundant & the relative pronoun "where" is not approriate here as "where" refers to place
(E) children born into a family
in which there are allergies will themselves probably develop an allergy
=> the use of "in which" is not correct as "in which" refers to place & "there are allergies" sounds awkward as if the allergies are not of family members.