I don't know a reference, but I think you can make the following argument.

The weighted projective stack $X$ is a $\mathbb{G}_m$-quotient of a scheme smooth over $R$, so it must be smooth over $R$ too: smoothness descends under flat morphisms, in the sense that if $Z \to W$ is faithfully flat and $Z$ is smooth, then so is $W$. (Here the map is a $\mathbb{G}_m$-torsor, so one doesn't need this stronger statement.)

For properness, you can use the valuative criterion (in the stacky version, as in Deligne-Mumford's paper on moduli of curves). Namely, assume that $R$ is a complete discrete valuation ring with residue field $K$. Then there are two things to show:

- The restriction from $R$-points of $X$ to $K$-points of $X$ is fully faithful (as a functor of groupoids).
- Given a $K$-point of $X$, after a ramified base extension to $K'$, it extends to an $R'$-point (where $R'$ is the integral closure in $K'$).

Over a local ring $R$, the groupoid of $X$-valued points is obtained by taking the set of tuples $\{x_0, \dots, x_n\}$ (at least one of which is invertible) and quotienting by the action of the
group $R^*$ (with the weighted action). There is no further sheafification needed since the Picard group is trivial. It follows that given two $R$-valued points, an isomorphism between the associated $K$-valued points necessarily comes from an isomorphism of $R$-valued points: the $K$-scalar inducing the isomorphism must be a unit.

Conversely, given a $K$-valued point $\{x_0, \dots, x_n\}$, then it is isomorphic to an $R$-valued point if the valuation of each $x_i$ is divisible by $a_i$, and after a ramified base change you can assume this. (This part, which uses the "stacky" version of the valuative criterion, is not necessary to run the argument for ordinary projective space.)