The authors investigate the existence of a random exponential attractor for the following nonautonomous stochastic Zakharov equations with multiplicative white noise on infinite lattices: \[ \begin{cases} \frac{1}{\lambda}\mathrm{d}\dot{x}_k +\alpha\mathrm{d}x_k+(A(x+|z|^2))_k \mathrm{d}t+\beta x_k\mathrm{d}t = f_k (t)\mathrm{d}t+ax_k \circ \mathrm{d}W^1, \\ \mathrm{i}\mathrm{d}z_k -((Az)_k+x_kz_k-\mathrm{i}\gamma z_k)\mathrm{d}t =h_k(t)\mathrm{d}t+bz_k \circ\mathrm{d}W^2, \\ x_k(\tau)=x_{k \tau}, \dot{x}_k(\tau)=x_{1,k \tau}, z_k(\tau)=z_{k \tau}, \end{cases} \] with \(t\geq \tau, k \in \mathbb{Z}, \tau \in \mathbb{R}\). Under some suitable assumptions, the authors provide an estimate of the solutions and prove that the continuous cocycle satisfies the Lipschitz continuity and the random squeezing property on a tempered random closed absorbing set. On the basis of these results, they prove the existence of a random exponential attractor.