Sure. I suspect the expense of new car batteries and the inconvenience of having them run down twice a day or so might make it a less optimal choice than just buying internet service.
Wifi routers consume on average about 6 watts. The new Volt has an 18.4 kWh battery, which could run such a wifi router for about 4 months. Or, put another way, running a wifi router all day would use about 0.005% of the Volt's battery capacity.
Obviously a lot of other support systems would be powered on as well, but still, the point is, you're not really going to notice the battery drain from a little wifi transmitter relative to the battery capacity needed to move a 3500 lb automobile 50+ miles.
Teslas will top up the 12V battery from the main battery even when the car is "off," but I don't know how other cars do it.
Unless there's an electric Silverado I don't know about?
//Disclaimer: I work for GM, and I worked in the factory where Hybrid Escalades were made.