It depends on the chickens, but a lot of them sit inside and stare at the snow, because they don't want to get cold feet by walking in it.
Some chickens do walk in the snow anyway, but they are not going to find any useful amount of food when it's buried in snow. If some places are free of snow, they might find stuff there (under an evergreen tree, inside a barn, etc.)
Some people shovel snow or put something on top of it (hay, straw, wood shavings, etc.) just so the chickens will come out and get exercise, but that doesn't really provide food for them.
If you already provide a complete chicken food, free-choice, you can just keep doing that in the winter. Make sure they have enough water, either by using a heated waterer, or carrying fresh water several times each day. The chickens may eat some snow, but you should not count on them getting any useful amount of water from it. Snow is mostly air, with only a little actual water in those fluffy crystals, and it takes a lot of body heat to melt it.
I've voted with respect to the Ex Battery hens I'm caring for
). Until two months ago my flock of four 6 year old hens and eight pullets/cockerel were kept exclusively in this enclosed space for safety. The predator exclusion fence I had wanted for years was finally completed in March, and ‘secure’ enough to allow them access by April. When the weather is agreeable and I am at home, the run door is open from morning to sunset, but with the hot temps here in TX., I shut them into their run around mid-day because they have a rudimentary evaporative cooling system of water pots, wet dirt and ceiling fans that make it a little bit less brutal for them. Forage here is sub-par, so I supplement their feed with fresh greens or veggie scraps and in the heat they get frozen watermelon on occasion. My chickens are treated very humanely, but I wouldn’t put them in the ‘pet’ category. Hope this info helps! 
