I think in most places the buyer of a vehicle still has to mail in a paper form (though if you buy from a car dealership, they'll take care of it), but at least in California, the seller can do their part of the transaction online (which is mainly to inform the DMV that someone else has the vehicle, so you won't be held responsible if something bad happens involving it).
And we don't have to register our moves at all; the government mostly doesn't care if we tell it where we live. Some agencies like the DMV do want to know our address so they can mail us a new driver's license or ID card when the old one expires (but these address changes and license renewals we can do online). The Postal Service will forward our mail to our new address for 3 months if we ask them to, but we don't have to if we don't care.
Obviously the government can and will eventually find out where you live if they need and want to, but there's generally no registration requirement.