I'm pretty sure that in the given example, it's a typo for "duty". I say this because if "duly" were really intended, "military" would have to be a noun. And that doesn't sound right in the phrase "the exigencies of military". Usually when "military" is a noun, it is preceded by "the". But
without having done everything that
the exigencies of military duty
require of him
makes perfect sense and sounds more natural. Moreover, "military duty" and "military duties" appear several times in the document, whereas "duly require" does not.
(On the other hand, the writers of this Ethiopian document may not be native English speakers, or their dialect may be a little different from mine.)
Aside from this example, I think the meaning of "duly require" is straightforward: it means to properly or fittingly require something. E.g. when I google (bing) for this phrase, the first hits contain
car financing companies duly
require a credit report...
Meaning, car financing companies require a credit report and that is proper/fitting/expected that they require one. This is opposed to a store unduly requiring your telephone number or social security number in order to complete a transaction... such a requirement does not seem justified.
So to answer your question, "duly require" is not quite the same as "require" or "need" (although someone could use it that way, i.e. with "duly" being redundant): it adds the sense that the requirement is proper or expected.
The sense of a possible redundancy I think comes from the fact that both "duly" and "require" convey a sense of obligation or "ought". But the "ought" applies to two different parties. If A duly requires something from B, the "duly" speaks of what is proper for A (namely, to require something), while the "require" puts an obligation on B, to provide something.