Acquiring digital representations of multivariate continuous-time (CT) signals is a challenge encountered in many signal processing systems. In practice, these signals are often obtained in order to extract some underlying information, i.e., for a specific task. Employing conventional task-agnostic analog-to-digital converters (ADCs) can be inefficient in such cases. In this work, we study task-based ADCs designed to obtain a digital representation of a multivariate CT input process to recover an underlying random parameter vector, referred to as the task. The proposed system employs analog filtering, uniform sampling, and scalar uniform quantization of the input process before recovering the task vector using a linear filter. We optimize the analog and digital filters and derive closed-form expressions for the achievable MSE in recovering a task vector from a set of bandlimited signals when utilizing a fixed quantizer resolution and sampling rate satisfying the Shannon-Nyquist sampling theorem. Guidelines for the design of practical acquisition systems are obtained from the structure of the MSE minimizing analog filter. Our numerical results, which consider the recovery of a set of matched filter outputs under a rate budget, demonstrate that the proposed approach substantially outperforms both, implementing the matched filter solely in the analog or digital domain.