In this sentence:
They did a great job orchestrating volunteers.
Is the gerund
orchestrating the object of the sentence?
orchestrating acts as a noun in the sentence, but I don't understand why it is the object of the sentence.
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This turn is the same as the following idiomatic one (OALD), except for the preposition "of" that is missing and the verb, which is a synonym.
make a good, bad, etc. job of something
It means "to do sth well, badly, etc". You can replace "make" with "do"; this latter is much more common than "make", which seems to be rarely used in comparison (ref.).
"They did a great job orchestrating volunteers." and "They did a great job of orchestrating volunteers." can be used to express the same meaning and this related form without "of", from all evidence, must be considered to be an idiom too. An examination of the books containing "did a good job" shows that the sense is the same; both "did a good job of XX…XXing" and "did a good job XX…XXing" are found and the contexts can be made out. It can also be deduced that instead of "of" one can use "on", as shown in OALD (synonym), and even "at", "with" and "in" (not recorded in OALD).
All these forms (They did a great job of/on/at/with orchestrating volunteers.) can be rendered by the more formal sentence below.
Considering after OALD that the locutions with "of" and "on" are idioms, there is no ground in a grammatical analysis and therefore no question of whether the gerund is an object or not.
It might be worthwhile spending some time comparing the present form with those in question in this query to the ELU. The opinion advanced in one of the answers is that the function of apposition is probably not what is relevant and that, most likely, one is dealing with an adverbial. A comparison should also help to come to the conclusion that "orchestrating …" is not an adverbial. This example (F line), which shows the same grammatical elements in the same order, is another example of a gerund phrase as an adverbial ("eating cakes" modifies the verb); "orchestrating …" does not modify the verb.
This is one of several cases in the English language where a preposition is elided. It can be analyzed as "a great job of orchestrating volunteers", with the part beginning "of" being a prepositional phrase. Even when "of" is omitted, it still is effectively the same as a prepositional phrase (though likely some folks would apply a different term).
A similar construction is "He's too smart a person to do something that dumb". Some argue that the "smart a person" construction actually predates "smart of a person", but the concept is the same, and it "a person" can be understood as if it was a prepositional phrase, beginning with "of".
Orchestrating is not the object of the sentence, nor is it a noun.
Orchestrating takes an object of its own, volunteers, thereby making clear that it is a verb and thus could not possibly be the object of the sentence. It is a non-finite clause that acts as a supplement.
Note orchestrating volunteers may be moved about in the sentence:
Orchestrating volunteers, they did a great job.
It may also take adjuncts of its own,
They did a great job speedily orchestrating volunteers.
There is no explicit indication of the semantic relation between the supplement orchestrating volunteers and the main clause. This has to be inferred from the content of the clauses. What is clear, however, is that its subject is controlled by the subject of the main clause: it was they who orchestrated volunteers.
They learned a lot orchestrating volunteers.
They were miserable orchestrating volunteers.
They got a great job orchestrating volunteers.
For the last one, orchestrating volunteers is a modifier in NP a great job orchestrating volunteers as the meaning would change were we to alter the order:
Orchestrating volunteers, they got a great job.
Here the natural interpretation would be while orchestrating volunteers...