There are a few techniques for people with injuries to get around in the real world; wheelchairs, crutches, and several other wheeled mobility devices come to mind.
Generally, if someone is dealing with a broken leg in winter, they would stay indoors, let a caretaker tend to them, and not travel on their own. However, I imagine the situation would change if the injury was relatively long-term (not healed after 4 weeks), and shoveling the roads to make flat paths every day was not a realistic option.
While snow is mostly considered an obstacle, if it's often relatively fresh and smooth - of the kind used for skiing/sledding, I imagine the option of relying on that smoothness to get around would become more likely, letting someone recovering from an injury retain some self-reliance. However, from what I've quickly researched no such device has been invented in real life - perhaps because even snowy communities tend to have snow cleared.
The thought that comes to mind would be something like a "sled-chair" someone could use to traverse a space the size of a spread-out village. Someone might carry a set of convertible crutches that can be used indoors, but also used as skiing poles to push their chair around. However, I'm trying to consider how practical it is if this person would have to handle the entire device's weight when approaching small inclines, with difficulty setting up any kind of effective braking system.
I'm specifically looking for ideas on what are the closest real-life inventions in the real world, as well as what are some unexpected concerns that would arise from use of such an invention.