Vegetation in Arctic and sub-Arctic bogs is not necessarily rotting. After a while the vegetation in a bog creates an acidic environment that preserves plant material and animal remains as though they were pickled in vinegar. That is why in Britain and Ireland there are extensive peat bogs many thousands of years old, and coal deposits which began in a similar way. In dryer places it is possible for plants and animals to be suddenly overwhelmed by winter conditions and to stay frozen for thousands of years, and these might generate CO2 and methane when they are frozen out.
A famous example of quick freezing is the bronze age warrior known as the Ice Man, who was found about 15 years ago frozen in the Italian Alps where he was overcome by a snow storm when he tried to cross in about 3,000 BC. So the answer is that some of this vegetation froze quickly, but most of it didn't, or at least was already preserved when it froze. It is possible that there have been brief episodes of thawing since then, so the freezing was not necessarily continuous. Some will decompose when it thaws completely, and some won't.