In this article, we present a new variational Bayes approach for solving the neuroelectromagnetic inverse problem arising in studies involving electroencephalography (EEG) and magnetoencephalography (MEG). This high-dimensional spatiotemporal estimation problem involves the recovery of time-varying neural activity at a large number of locations within the brain, from electromagnetic signals recorded at a relatively small number of external locations on or near the scalp. Framing this problem within the context of spatial variable selection for an underdetermined functional linear model, we propose a spatial mixture formulation where the profile of electrical activity within the brain is represented through location-specific spike-and-slab priors based on a spatial logistic specification. The prior specification accommodates spatial clustering in brain activation, while also allowing for the inclusion of auxiliary information derived from alternative imaging modalities, such as functional magnetic resonance imaging (fMRI). We develop a variational Bayes approach for computing estimates of neural source activity, and incorporate a nonparametric bootstrap for interval estimation. The proposed methodology is compared with several alternative approaches through simulation studies, and is applied to the analysis of a multimodal neuroimaging study examining the neural response to face perception using EEG, MEG, and fMRI.