In sound field control applications, it is commonly assumed that one has access to an accurate representation of the sound field in the region of interest. This is a problematic assumption since the reconstruction of a sound field from available microphone measurements is especially challenging in real-time applications where only causal measurements are available. Notably, causal time-windowed observations introduce correlation between frequency components, making sound field reconstruction methods that process each frequency band independently sub-optimal. In this work, we formulate a causal finite-window spatio-temporal linear minimum mean-square error estimator for sound field reconstruction. The sound field is modeled as the solution to the wave equation driven by a stationary stochastic spatio-temporal source distribution, which induces a physically interpretable covariance function. It is shown that this covariance function is closely related to the classical diffuse-field coherence model. Since the computational complexity grows rapidly with the number of spatio-temporal observations, we formulate a budget-constrained spatio-temporal sample selection approach to minimize the posterior reconstruction variance. The proposed estimator and sampling strategy are evaluated using both simulated and measured sound fields, demonstrating improved short-window reconstruction compared to frequency domain finite-window baselines.