This paper presents the development and evaluation of an optimization-based autonomous trajectory planning algorithm for the asteroid reconnaissance phase of a deep-space exploration mission. The reconnaissance phase is a low-altitude flyby to collect detailed information around a potential landing site. Although such autonomous deep-space exploration missions have garnered considerable interest recently, state-of-the-practice in trajectory design involves a time-intensive ground-based open-loop process that forward propagates multiple trajectories with a range of initial conditions and parameters to account for uncertainties in spacecraft knowledge and actuation. In this work, we introduce a stochastic trajectory optimization-based approach to generate trajectories that satisfy both the mission and spacecraft safety constraints during the reconnaissance phase of the Deep-space Autonomous Robotic Explorer (DARE) mission concept, which seeks to travel to and explore a near-Earth object autonomously, with minimal ground intervention. We first use the Multi-Spacecraft Concept and Autonomy Tool (MuSCAT) simulation framework to rigorously validate the underlying modeling assumptions for our trajectory planner and then propose a method to transform this stochastic optimal control problem into a deterministic one tailored for use with an off-the-shelf nonlinear solver. Finally, we demonstrate the efficacy of our proposed algorithmic approach through extensive numerical experiments and show that it outperforms the state-of-the-practice benchmark used for representative missions.