The section 12.2  of Huang's treatise on Statistical Mechanics (1963) gives hints of possible connections between the Ising model and Clifford algebras, but of course $\textit{under the physicist point of view}$. Below I summarize the solution of the model without magnetic field, without proof and proper mathematical definitions. This is $\textit{not}$ an answer (Ceci n'est pas une pip), but maybe you can identify some element towards a rigorous and clear mathematical formulation.

The partition function of the Ising model on a squared lattice with
$n^{2}$ spins, without external field, with isotropic exchange coupling
$\epsilon$, is
\begin{equation}
Z(\beta\epsilon)=\text{Tr}\mathsf{\:{P}}^{n},
\end{equation}

with the $2^{n}\times2^{n}$ matrix $\mathsf{{P}}$ given by 

\begin{equation}
\mathsf{{P}}=\left(2\sinh\left(2\beta\epsilon\right)\right)^{n/2}\left[\frac{1}{2}\left(1+\mathsf{{U}}\right)\mathsf{{V}}^{+}+\frac{1}{2}\left(1-\mathsf{{U}}\right)\mathsf{{V}}^{-}\right],
\end{equation}

being $\beta$ the inverse temperature, and
\begin{align}
\mathsf{{V}}^{\pm} & =e^{\pm i\phi\Gamma_{1}\Gamma_{2n}}\left[\prod_{\alpha=1}^{n-1}e^{-i\phi\Gamma_{2\alpha+1}\Gamma_{2\alpha}}\right]\left[\prod_{\lambda=1}^{n}e^{-i\theta\Gamma_{2\lambda}\Gamma_{2\lambda-1}}\right]\\
\mathsf{{U}} & =i^{n}\Gamma_{1}\Gamma_{2}\cdots\Gamma_{2n},
\end{align}

where $\phi=\beta\epsilon$, $\tanh\theta=e^{-2\phi}$, and the $2n$
matrices $\Gamma_{\mu}$ $\left(\mu=1,\cdots,2n\right)$ (-of size
$2^{n}\times2^{n}$) defined by the anticommutation rule
\begin{equation}
\Gamma_{\mu}\Gamma_{\nu}+\Gamma_{\nu}\Gamma_{\mu}=2\delta_{\mu\nu}.
\end{equation}