It is known that there exists a fine moduli space for marked (nonalgebraic) K3 surfaces over $\mathbb{C}$. See for example the book by Barth, Hulek, Peters and Van de Ven, section VIII.12. Of course the marking here is necessary, otherwise the presence of automorphisms can be used to construct isotrivial non trivial families of K3.

Assume that we want to construct an algebraic analogue of this. Of course some structure has to be added; I am thinking of something like torsion points for elliptic curves.

Is there a way to add some structure and actually build a fine moduli space of (K3 + structure) which is defined over $\mathbb{Z}$?

Of course this will only parametrize algebraic K3 surfaces, so it will need to have infinitely many components. A second question, if we do not want to add structure, is

Is there a fine moduli stack of algebraic K3 surfaces which is algebraic (either in the DM or Artin sense)?

It is not difficult to produce K3 surfaces with a denumerable infinity of automorphisms, so I'm not really expecting the answer to be yes, but who knows.

Edit: since there seems to be some confusion in the answers, the point is that I'm asking for a FINE moduli space. I'm aware that one can consider moduli spaces of polarized K3, and this is why I wrote "Of course this will only parametrize algebraic K3 surfaces, so it will need to have infinitely many components."