Ok, first a disclaimer. I've only had chickens in our coop for a week and a half now, so when it comes to chicken things i'm a novice. Also, I was unfamiliar with vinyl siding, and was quite surprised to find that it's one of the most commonly used outer materials for houses in Northern America (at least according to Wikipedia).
But here are my thoughts on insulation, that I do have some experiences with, living in a quite cold climate.
I would assume that vinyl siding nailed on top of your framing really doesn't breathe all that much, so from a ventilation aspect you're probably not gonna reduce the airflow if you do insulate the walls. Most stuff I would use to insulate with would require a bit of a breathing space between the outer material and the insulation though, so you should study up on installation instructions for whatever insulation, if any, you choose to go with.
The benefit on insulating structures is that you can reduce the impact of quick changes in temperature, the sun wont heat up the coop as much, and during colder times, a properly insulated structure requires a lot less heating. Also, putting in insulation now is a lot easier than doing it later.
Personally I went with plywood on the inside (which lets some moisture through), rock wool as an insulator (which lets moisture pass through even more easily), a sheet of windproof paper (which also let's moisture through) and then a 20mm breathing space which let's the structure dry out, and then wood siding on the outside, which of course is painted. The outer walls are protected from moisture by the roof, so even if they do get a bit wet on the outside during windy days, they are not subjected to that much direct rain.
You still want to ventilate the inside of the coop quite well, because that's going to be the main source of moisture.
I don't now how houses are usually built in the States, but at least in a Finnish climate this kind of constructions work well. Last summer we redid the outer boards on our cottage which had been standing 30 feet from the sea for 60 years. It had a similar structure, and there were absolutely no signs of rot noticeable in the walls.