Forecasting tasks using large datasets gathering thousands of heterogeneous time series is a crucial statistical problem in numerous sectors. The main challenge is to model a rich variety of time series, leverage any available external signals and provide sharp predictions with statistical guarantees. In this work, we propose a new forecasting model that combines discrete state space hidden Markov models with recent neural network architectures and training procedures inspired by vector quantized variational autoencoders. We introduce a variational discrete posterior distribution of the latent states given the observations and a two-stage training procedure to alternatively train the parameters of the latent states and of the emission distributions. By learning a collection of emission laws and temporarily activating them depending on the hidden process dynamics, the proposed method allows to explore large datasets and leverage available external signals. We assess the performance of the proposed method using several datasets and show that it outperforms other state-of-the-art solutions.