The Soviets implemented a "scorched earth" policy designed to deprive the occupying Germans of anything and everything they might use to sustain themselves. This included burning down all usable buildings and their contents, killing any livestock that could not be brought east, poisoning wells and food, destroying machinery, burning oil fields and crops, ect. From what I understand although the Germans were not properly clothed for winter, once this problem became strikingly apparent this was attempted to be remedied by salvaging Russian clothes like you said, military issued winter wear, but also through clothing drives back home, where German families would send winter clothes and blankets to those deployed. However and getting back to "scorched earth", no matter how many pairs of socks or coats you have you need more then that to survive a long Russian winter of living, traveling, and fighting outside. You need shelter, downtime to rest (sweating can kill you), warm food, dry clothes to change into (harder to salvage), oils to keep your weapons and machines from seizing up in the cold (as well as spare parts), and most importantly fuel. This last bit was most scarce, with Panzer divisions and units of mechanized infantry running out of fuel throughout the campaign, and a reason why it was imperative for the Germans to strike south and capture the Baku oilfields in the Caucuses. This obviously did not happen, as the advance was stopped at Stalingrad.

While it might have been theoretically possible to supply the troops with all these necessities, the Germans treated occupied populations as sub-humans. This deprived them of potential assistance (Soviet occupied people like the Ukrainians initially welcomed the Germans as liberators, but soon turned against them for obvious reasons) and created (along with the rapid German advance that left behind many Red Army units) partisan resistance that harried the rear and made supply, rest, and downtime from the front, very difficult.