We study a disordered mean field network of purely excitatory leaky integrate-and-fire neurons with short-term synaptic plasticity. If the synaptic resources display a fast dacay, the dynamical phase diagram becomes especially interesting. In this case indeed, the quasi-synchronous and the asynchronous phases are separated by a nontrivial, collective, bursty regime with avalanches. In the homogeneous case without disorder, the system synchronizes and the bursty behavior is reflected by the presence of sychronous chaos described by a two dimensional chaotic discrete map. Numerical simulations show that the bursty chaotic phase with avalanches exhibits the spontaneous emergence of persistent time correlations. Our analysis reveals a mechanism for the generation of irregular avalanches that emerges from the combination of disorder and the deterministic underlying chaotic dynamics.