I will describe experimental progress towards demonstrating a newly proposed hardware architecture and protocol for connecting many local quantum processors contained within an optical cavity. The scheme is compatible with trapped ions or Rydberg arrays and realizes teleported gates between any two qubits by distributing entanglement via single-photon transfers through a cavity. In contrast to previous proposals for quantum computing with optical cavities, we employ heralding to achieve high-fidelity entanglement even with a cavity of moderate quality. For processors composed of trapped ions in a linear chain, a single cavity with realistic parameters successfully transfers photons every few μs, enabling the any-to-any entanglement of 20 ion chains containing a total of 500 qubits in 200 μs with both fidelities and rates limited only by local operations and ion readout. My experimental efforts are focused on processors composed of Rydberg atoms, where our method fully connects a large array of thousands of neutral atoms. The connectivity afforded by our architecture is extendable to tens of thousands of qubits using multiple overlapping cavities, expanding capabilities for NISQ era algorithms and Hamiltonian simulations, as well as enabling more robust high-dimensional error-correcting schemes.