The construction of synthetic gene circuits requires the rational combination of multiple regulatory components, but predicting their behavior can be challenging due to poorly understood component interactions and unexpected emergent behaviors. In eukaryotes, chromatin regulators (CRs) are essential regulatory components that orchestrate gene expression. Here, we develop a screening platform to investigate the impact of CR pairs on transcriptional activity in yeast. We construct a combinatorial library consisting of over 1,900 CR pairs and use a high-throughput workflow to characterize the impact of CR co-recruitment on gene expression. We recapitulate known interactions and discover several instances of CR pairs with emergent behaviors. We also demonstrate that supervised machine learning models trained with low-dimensional amino acid embeddings accurately predict the impact of CR co-recruitment on transcriptional activity. This work introduces a scalable platform and machine learning approach that can be used to study how networks of regulatory components impact gene expression.