When different SC answers take a verb and change it into different parts of speech, that sometimes signals an idiom issue, and sometime signals a parallelism issue.
In this sentence, the word "work" is used as a verb in (A), a noun in (B) and (D), a modifier in (C), and an infinitive in (E). Although there is a parallel marker in this sentence (the word "and"), parallelism doesn't help us to choose among these verb forms. So let's treat it as an idiom question.
Usually, some verb earlier in the sentence will determine the correct idiom. In this case it is the verb "order." To choose the correct idiom, you will have to rely on your ear. Your ear may work better with a simpler sentence, so let's consider these options:
(A) He ordered them should work.
(B) He ordered them would work.
(C) He ordered them working.
(D) He ordered them the work.
(E) He ordered them to work.
Only E is idiomatic.
By the way, the
Manhattan GMAT Sentence Correction Strategy Guide contains a list of the correct forms of many common idioms. If you have the Fourth Edition of that book, you can find an entry for "order" at the top of page 162 (you'll see that we agree with Chembeti, or perhaps that he agrees with us).
Saikarthikreddy's remarks are interesting and mostly correct, but some of that analysis is too much for the time constraints of the GMAT, and some isn't 100% right ("working" isn't a verb in C, the subjunctive is used more broadly than your remarks on B suggest). If the answers force you to choose among verb forms, or among prepositions, you may have to rely on your ear.