Can someone explain to me the role of "chur" in this sentence?
It's the verbal noun of cuir, the verb for "put". Here, it's just another way of expressing "call" (Put a call on someone).
Does the use of 'chur' (put) infer that a request is being made of the people? as in to say, some kind of responsibilty is being put on them to act?
No — glaoch a chur might be better interpreted literally as “to place a call” in English.
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I thought that "chun" requires the genitive?
Should this not be "Táim chun glaoigh a chur ar na daoine"?
In most cases chun does take the genitive. However, expressions of intention, as here, are an exception, in which it usually takes the common (nominative) form.
I note that GnaG hedges its bets slightly on that point - "after chun here, mostly no genitive", and provides the example of "chun an fhírinne a rá", but the FGB offers both the nominative and genitive versions of that phrase - "chun an fhírinne, na fírinne, a rá", "to tell the truth".
So it sounds like "Táim chun glaoigh a chur ar na daoine" wouldn't necessarily be wrong (at least for some Irish speakers), but "Táim chun glaoch a chur ar na daoine" is not a problem.
What underlies this is the functional genitive, where a noun functions as a genitive but remains in the common/nominative form.
There are a number of cases where this happens, one of which is the subject or object of a verbal noun in the construction subject/object + a + verbal noun. In "chun glaoch a chur", glaoch functions as a genitive, satisfying chun's need for the genitive, but retains the common form.
Another case is a verbal noun that directly follows chun when expressing purpose, e.g. "Táim chun glaoch ar na daoine". There are exceptions to this case that may explain why GnaG seems to hedge.
This is covered well in the Christian Brothers' grammar books. I don't see why FGB gives "chun na fírinne a rá", but even if glaoigh could be correct here, I would want a good reason (such as dialect) to go against the advice of the Christian Brothers' grammar. Therefore, I think glaoch is at least preferable.
To me "about to" expresses that something is imminent, which would usually be expressed using ar tí: "Táim ar tí glaoch a chur ar na daoine."
As noted elsewhere in this discussion, chun expresses intention here, which "going to" can express in English.