Can you install a dog fence in winter, when the ground is frozen?

The answer is a yes and a no.  Your dog fence will work just fine in winter when the ground is frozen.  But, it can be very difficult to bury the boundary wire when the ground is frozen solid.  Even, a trencher can have difficulty digging through ice and frozen ground.

One trick that can sidestep this difficulty is to use ground staples to pin the wire to the frozen ground, thereby avoiding the need to dig.  Once the ground starts to thaw (assuming you don’t have permafrost), then you can use a trencher to bury the wire.