The vanishing inertia of synchronous generators in transmission systems requires the utilization of renewables for inertial support. These are often connected to the distribution system and their support should be coordinated to avoid violation of grid limits. To this end, this paper presents the Physics-informed Actor-Critic (PI-AC) algorithm for coordination of Virtual Inertia (VI) from renewable Inverter-based Resources (IBRs) in power distribution systems. Acquiring a model of the distribution grid can be difficult, since certain parts are often unknown or the parameters are highly uncertain. To favor model-free coordination, Reinforcement Learning (RL) methods can be employed, necessitating a substantial level of training beforehand. The PI-AC is a RL algorithm that integrates the physical behavior of the power system into the Actor-Critic (AC) approach in order to achieve faster learning. To this end, we regularize the loss function with an aggregated power system dynamics model based on the swing equation. Throughout this paper, we explore the PI-AC functionality in a case study with the CIGRE 14-bus and IEEE 37-bus power distribution system in various grid settings. The PI-AC is able to achieve better rewards and faster learning than the exclusively data-driven AC algorithm and the metaheuristic Genetic Algorithm (GA).