To address the difficulty of simultaneously ensuring operational security and delivery reliability in flexibility service scheduling of distribution networks under multiple sources of uncertainty, an optimal scheduling method based on distributionally robust chance-constrained programming is proposed. A hierarchical model incorporating hard security constraints and distributionally robust chance constraints is established, in which the hard security constraints are used to impose operational security requirements on nodal voltage, branch capacity, and energy boundaries of energy storage within the prescribed uncertainty support and under the adopted model conditions, while the distributionally robust chance constraints are introduced to characterize the delivery risk of active and reactive flexibility services at the point of common coupling. Furthermore, linear decision rules are employed to describe real-time adjustment strategies, thereby enabling the coordinated optimal scheduling of controllable loads, energy storage, and reactive power resources of photovoltaic inverters. The resulting model is then reformulated as a second-order cone programming problem for efficient solution. Case studies on the IEEE 33-bus and IEEE 123-bus systems demonstrate that the proposed method can improve the delivery reliability of flexibility services and reduce out-of-sample violation risk while ensuring the secure operation of the distribution network, with satisfactory economic performance and engineering applicability.