A time-dependent analysis and spectral characterization of the plate's response under stochastic excitation are presented. The two-dimensional structural element is modeled by accounting for two key aspects: the rheological time-dependent stress–strain behavior of the material and the presence of long-range nonlocal interactions. This investigation may provide valuable insights for the design of nano- and micro-scale devices—such as generators, piezoelectric systems, NEMS, and MEMS—as well as dielectric elastomeric plates (e.g., components of ocean wave energy harvesters). Among the available nonlocal models, we adopt the stress-driven approach, which describes the nonlocal strain as a convolution integral of the entire local strain field weighted by a suitable averaging kernel. With an appropriate choice of this kernel, the integral model leads to a well-posed differential formulation that is more tractable from a computational standpoint. Furthermore, time-dependent stress–strain relations based on versatile fractional-differential models are implemented to introduce a constitutive law capable of capturing both nonlocal effects and rheological behavior. The resulting nonlocal viscoelastic stress–strain relation is combined with the Kirchhoff plate kinematic model under the assumption of axisymmetry to derive the dynamic equation of motion, considering a Gaussian stochastic process as the external input. Mode shapes, frequency analysis, power spectral density, and both stationary and non-stationary variances of the displacement response are presented and discussed. The results highlight the influence of factors such as geometry, nonlocal parameters, and viscoelastic coefficients on the mechanical response and the dominant frequencies of the structure.