Realistically we wouldn’t just do this overnight. You would take a couple of years to work all the details out. Peaceful dissolution of nations has happened many times before. You don’t just shut down the old empire like turning off a light switch. But there would be existing institutions to build off of. The existing state national guards could be expanded to serve as full military units.

As far as doing this Constitutionally, the process is a bit dubious. But really, it doesn’t matter. If you’re at the point of the population being willing to voluntarily dissolve the country, you simply ignore the old constitution entirely. If a president and Congress were elected with a mandate of dissolving the US entirely, they could simply do it and there wouldn’t be anyone to stop them. Hell, you could probably do this just by electing a president on a platform of dissolving the country. Yes, it really wouldn’t be constitutionally valid, but in these kind of situations, that’s not really relevant anymore.