Existing research has either adapted the Probably Approximately Correct (PAC) Bayesian framework for federated learning (FL) or used information-theoretic PAC-Bayesian bounds while introducing their theorems, but few consider the non-IID challenges in FL. Our work presents the first non-vacuous federated PAC-Bayesian bound tailored for non-IID local data. This bound assumes unique prior knowledge for each client and variable aggregation weights. We also introduce an objective function and an innovative Gibbs-based algorithm for the optimization of the derived bound. The results are validated on real-world datasets.