For a range of scenarios arising in sensor networks, control and edge computing, communication is event-triggered; that is, in response to the environment of the communicating devices. A key feature of device activity in this setting is correlation, which is particularly relevant for sensing of physical phenomena such as earthquakes or flooding. Such correlation introduces a new challenge in the design of resource allocation and scheduling for random access that aim to maximize throughput or expected sum-rate, which do not admit a closed-form expression. In this letter, we develop stochastic resource optimization algorithms to design a random access scheme that provably converge with probability one to locally optimal solutions of the throughput and the sum-rate. A key feature of the stochastic optimization algorithm is that the number of parameters that need to be estimated grows at most linearly in the number of devices. We show via simulations that our algorithms outperform existing approaches in terms of the expected sum-rate by up to 30% for a moderate number of available slots.