Decomposition will greatly slow down in Winter but it won't completely stop. If you have organic matter you can put it in on the soil at anytime, but if you are going to buy organic matter, I would wait until Spring.
Organic matter is always decomposing in soil. The amount that remains year-to-year (without addition) is dependent on climate and vegetation. Basically we add more organic matter to give our gardens a boost, especially during the growing season. While the added organic matter is still present, it will improve drainage in heavy soil, water retention in light soil, and will feed the microbiota in all soils. Even after most of the added organic matter has turned into CO2, there will still be a nutrient remnant that will enrich the soil.
Organic mulches will also eventually work into the soil, but while on the surface they will help conserve water, prevent erosion, and suppress weeds. This is one of several reasons I'm against plastic weed cloth. It acts as barrier preventing organic mulch from being fully incorporated into the soil below. By the way, the word 'mulch' does not refer any particular material, it refers to covering the soil surface with any such material, be it wood chips, bark, leaf mold, rocks, pebbles, etc.
Adding organic matter year after year will slowly improve our soils, but erosion, crop harvesting, and other removal of plant matter will work against this. That is why it is great to compost on-site as much garden waste as one can. It is best to do this before buying it by the bagful or truck load, though of course, most avid gardeners will do both sooner or later.
I know, this a long post, and you may already know much of this, but I thought it would be helpful to set it down for other readers as well.