Obtaining digital representations of multivariate continuous-time (CT) signals is a challenge encountered in many signal processing systems. In practice, these signals are often acquired to extract some underlying information, i.e., for a specific task. Employing conventional task-agnostic analog-to-digital converters (ADCs), typically designed to minimize the mean squared error (MSE) in reconstructing the CT input signal, can be costly and energy-inefficient in such cases. In this work, we study task-based ADCs, which are designed to obtain a digital representation of a multivariate CT input process with the goal of recovering an underlying statistically related 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 digital recovery filter. We optimize the analog and digital filters and derive closed-form expressions for the achievable MSE in recovering the task vector from a set of analog signals when utilizing ADCs with a fixed sampling rate and amplitude resolution. Based on our derivation, we provide guidelines for designing practical acquisition systems subject to a constraint on the bit rate. Our analysis proves that the intuitive approaches of either recovering the task vector solely in digital or designing the analog filter to estimate the task vector are inferior to the proposed joint design. We then consider the recovery of a set of matched filter outputs under a rate budget. We numerically verify our theoretical observations and demonstrate that task-based ADCs substantially outperform analog matched filtering as well as applying the matched filter solely in the digital domain. When acquiring signals for a task under tight bit budgets, we also show that it is often preferable to sample below the Nyquist rate instead of reducing the quantization resolution.