The theory of $(\varphi, \Gamma)$-modules works for $\mathbf{Z}_p$-linear representations, but one has to use a slightly different coefficient ring $\mathbf{A}_K$. This is explained very clearly in section 1.4 of the first Cherbonnier--Colmez paper ("Representations $p$-adiques surconvergentes") which you cite in your question. This goes all the way back to Fontaine's work in the late 80's, and thus pre-dates the Robba ring theory by about 10 years.

The difficulty is that $\mathbf{A}_K$ does not embed in the Robba ring. There is a subring $\mathbf{A}_K^\dagger \subset \mathbf{A}_K$ which does embed in $\mathcal{R}_K$, and the main result of Cherbonnier--Colmez is that every etale $(\varphi, \Gamma)$-module over $\mathbf{A}_K$ can be descended to $\mathbf{A}_K^\dagger$. But $\mathbf{A}_K^\dagger$ and its cousin $\mathbf{B}_K^\dagger = \mathbf{A}_K^\dagger[1/p]$ are both much smaller than the Robba ring, and the Robba ring does not have any natural integral subring --there is no nice ring $A \subset \mathcal{R}_K$ such that $\mathcal{R}_K = A[1/p]$ and $A$ is $p$-adically separated.