If you view a $\mathbb{Z}$-valued (Cartier) divisor (on say, an integral separated scheme X) as a $\mathbb{G}\_m$-torsor on X with generic trivialization, then for any torus T with character group $X^\ast(T)$, an $X^\ast(T)$-valued divisor on X is a T-torsor with generic trivialization. This shows up for X a curve, e.g., in some treatments of geometric class field theory and Gaitsgory's Twisted Whittaker model paper. The analogous construction will work for any group of multiplicative type, replacing the character group with a suitable fpqc sheaf of abelian groups on the base.
I don't know what you get if you want the action of a larger ring - you can certainly tensor up the character groups as abelian sheaves, so if you're looking for tori with some kind of CM, I guess that will work.