I think it's less "trade" and more like, the way some monasteries in the real world are supported by their local populace, so too were the Air Temples. They no doubt farm what they can up there, but the bulk of their food is probably donations from the Earth Kingdom and Fire Nation villages nearest each temple. And in return, the Air Nomads would give spiritual guidance and assistance. None of this is ever explicitly addressed but I feel like it is implied especially in FC Yee's novels. Keep in mind you can get pretty much anywhere in the world from any air temple in a matter of days by sky bison so collecting donations would be pretty easy.

I feel like I recall the show making some mention of those monasteries not being occupied year-round.

They are occupied year-round but not by all the same people necessarily. Children are born in the East and West Temples (the ones where nuns take priority) and the boys once they get old enough ~~(probably once weaned)~~ (edit: I just remembered this is wrong, the cutoff point is once they bond with a sky bison. The sky bison are also raised at the East and West temples so boys can leave once they've befriended "their" sky bison which seems to happen around age 3 or 4) are taken to the North and South temples (where monks take priority) and they seem to stay largely at one temple which they then consider their "home" temple. Then once they're old enough not to need a direct guardian looking after them, they're free to wander the world and stop by any of the four temples as they please. But they're still considered residents of one temple and visitors to any of the other temples.