Contrary to the feminists’ hope that the advent of IT sector and the wide acceptability of equal pay for equal work has the ability to soon to achieve it, social equality is still a distant dream for most women.
(A) has the ability to soon to achieve it,
Usage of has for plural subject "advent of IT sector
and the wide acceptability of ..."is wrong here.
(B) could soon achieve it
No issues as such. Should be a correct choice.
(C) would be capable of its achievement soon
Now
its here refer back to the subject "advent of IT sector and the wide acceptability of ..." which is wrong both in terms of meaning and Subject Verb relationship.
(D) to be able to soon achieve it
Here verb is missing . This is something like "Subject to be able to soon achieve it". There is no verb here.
(E) soon achieving it
Here also verb is missing. This is something like "Subject soon achieving it". We have to have a helping verb such as is, are for this sentence to make some sense.