There are some weeping mulberry trees by me that I can take cuttings off of, they're nice because most of the berries are protected from bird predation, unlike my dwarf mulberries.
I've read that weeping mulberries are a regular mulberry with a weeping mulberry grafted onto it.

What do they do if they're not grafted to a root stock just sprawl on the ground like a limp noodle or do they grow a trunk?

I don't know what would happen, but I do know that it's pretty easy to get ahold of rootstock for mulberries if you want to just graft one. I think they usually are in the form of Russian Mulberry seedlings. I just checked Burnt Ridge, for example; they sell them right now for $3.50