We consider an ad-hoc network of wireless sensors that harvest energy from the environment and broadcasts measurements independently, at random, provided sufficient energy is available. Clients arriving at the network are interested in retrieving measurements from an arbitrary set of sensors of some fixed size s. We show that the sensors broadcast measurements according to a phase-type distribution. We determine the probability distribution of the time needed for a client to retrieve s sensor measurements. We provide a closed-form expression for the retrieval time of s sensor measurements for an asymptotically large capacity of the sensor battery or the rate at which energy is harvested. We also analyze numerically the retrieval time of s sensor measurements under various assumptions regarding the battery capacity of the sensors, the energy harvesting and consumption processes. The results provide a lower bound for the energy storage capacity of the sensors for which the retrieval time of measurements is below a targeted level. It is also shown that the ratio between the energy harvesting rate and the broadcasting rate significantly influences the retrieval time of measurements, whereas deploying sensors with large batteries does not significantly reduce the retrieval time of measurements. Numerical experiments also indicate that our theoretical results generalize to non-identical energy harvesting rates, various amount of energy consumed upon a broadcast and non-exponential distributions of the energy harvesting and broadcasting processes.