Growing penetrations of distributed energy resources (DERs) in distribution systems have motivated the design of controllers that leverage DER capabilities to achieve system-wide objectives. These controllers may be either grid-agnostic or grid-aware, depending on whether distribution network constraints are considered. Grid-agnostic controllers have the benefit of not requiring network models or system measurements, but may cause dangerous constraint violations. Rather than develop a specific controller, this paper considers the potential impacts of DER controllers with respect to network constraint violations. Specifically, this paper develops an optimization-based method to rigorously certify when any grid-agnostic controller can be applied without concern regarding network constraint violations, or, conversely, when grid-aware control may be needed to maintain distribution grid security. The proposed method uses convex optimization techniques to bound the impacts of load variability, given a subset of buses with voltage measurements and control. The method either provides a certificate for secure operation or identifies potentially critical constraints and the need for additional controllability. Numerical tests illustrate the ability to certify secure operation for different ranges of variability.