Delivering timely status information of a random process has become increasingly important for time-sensitive applications, e.g., vehicle tracking and environment monitoring. We consider an IoT sensing network, where a cache-enabled wireless edge node receives on-demand requests from multiple users to send status updates on physical quantities, each measured by an energy harvesting sensor. To serve users' requests, the edge node uses the current information state (i.e., the number of requests, battery level, and AoI for each sensor) to decide whether to command a sensor to send a status update or to retrieve the most recently received sensor's measurements from the cache. We aim at finding the best actions of the edge node to minimize the average AoI of the served measurements at the users, i.e., average on-demand AoI. We model this as a Markov decision process problem and derive a relative value iteration algorithm to find an optimal policy. Simulation results illustrate the threshold-based structure of an optimal policy and show that the proposed on-demand updating policy outperforms the greedy (myopic) policy and also, by accounting for the per-sensor request frequencies and intensities, the pure average AoI minimization policy that keeps the edge node updated regardless of requests.