Solving constraint satisfaction problems (CSPs) is a notoriously expensive computational task. Recently, it has been proposed that efficient stochastic solvers can be obtained through appropriately configured spiking neural networks performing Markov Chain Monte Carlo (MCMC) sampling. The possibility to run such models on massively parallel, low-power neuromorphic hardware holds great promise; however, previously proposed networks a re based on probabilistically spiking neurons, and thus rely on random number generators or external noise sources to achieve the necessary stochasticity, leading to significant overhead in the implementation. Here we show how stochasticity can be achieved by implementing deterministic models of integrate and fire neurons using subthreshold analog circuits that are affected by thermal noise. We present an efficient implementation of spike-based CSP solvers using a reconfigurable neural network VLSI device, and the device's intrinsic noise as a source of randomness. To illustrate the overall concept, we implement a generic Sudoku solver based on our approach and demonstrate its operation. We establish a link between the neuron parameters and the system dynamics, allowing for a simple temperature control mechanism.