The section 12.2 of Huang's treatise on Statistical Mechanics (1963) gives hints of possible connections between the Ising model and Clifford algebras. 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 elements 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}