HVAC air handler with a built-in humidifier
Whole house humidifiers exist, and could be wired to work in your situation. This would require the air handler blower to run during the winter, and humidity level detection might be a challenge.

Typically the humidifier is installed on the return ducting of the air handler, and a bypass duct is connected to allow humidity to be added to the air. A humidistat is also installed on the return air ducting, which allows the humidifier to activate based on the humidity in the air coming from the house. Since you won't want the air handler to run constantly, you'll have to figure a way to passively detect the humidity level throughout the house. You'll then activate the blower, and the humidifier based on that.