B uses the proper pronoun.
Others have highlighted errors.

The typical scanning for splits and elimination strategy makes this question extremely simple to answer in least time.

Meaning analysis

Gentleman (or Gentlemen) of the 18th century protected his/their clothing by poking his/their head through a device that resembled the stocks. The protection for their clothing was required when someone powdered their wig.

Meaning not really required here, as we notice it is a simple case of singular/plural noun/pronoun error.

Scan 1:

Gentleman v/s Gentlemen (3:2 split). Wherever plural "gentlemen" is used, the corresponding pronouns should be "...their clothing", "...their wigs", and "their heads".
Basis this, eliminate D and E.

Scan 2:

Conversely, wherever singular "gentleman" is used, the corresponding pronouns should be "...his clothing", "...his wig", and "his head".
Basis this, eliminate A and C.

What are we left with?

Voila! B! "gentleman" .... "his clothing" ...."his wig" ...."his head". All singular. Correct option.
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This question has the error of using a wrong pronoun and also using a singular noun where a plural noun is necessary.
The subject of the given sentence is ‘well dressed-gentleman’, which is a singular noun.

So, the use of the plural pronoun in the phrase ‘their wig’ is grammatically wrong. In addition, the phrases ‘their wig’ and ‘their head’ implies that many of them together had only one wig and only one head, which is absurd. So, (A) is wrong.

(C) also incorporates the first error identified above, and can be eliminated.

(D) and (E) both have the same error in reverse, with the plural noun ‘gentlemen’ being represented by the singular pronoun ‘his’ in the latter part of the sentence.

(B) corrects these errors and results in a clear and grammatical statement. Therefore (B) should be the answer.

A) gentleman of the eighteenth century protected their clothing while having their wig powdered by poking their head
(B) gentleman of the eighteenth century protected his clothing while having his wig powdered by poking his head
(C) gentleman of the eighteenth century protected their clothing while having their wigs powdered by poking their heads
(D) gentlemen of the eighteenth century protected his clothing while having his wig powdered by poking his head
(E) gentlemen of the eighteenth century protected their clothing while having his wig powdered by poking his head

What is the difference between B and D?

gentleman =/= gentlemen

One is a singular subject, the other is plural.