Recently, graph neural architecture search (GNAS) frameworks have been successfully used to automatically design the optimal neural architectures for many problems such as node classification and graph classification. In the existing GNAS frameworks, the designed graph neural network (GNN) architectures learn the representation of homogenous graphs with one type of relationship connecting two nodes. However, multi-view graphs, where each view represents a type of relationship among nodes, are ubiquitous in the real world. The traditional GNAS frameworks learn the graph representation without considering the interactions between nodes and multiple relationships, so they fail to solve multi-view graph-based problems, such as multi-view graphs modelling the biomedical entity and relation extraction tasks. In this paper, we propose MVGNAS, a multi-view graph neural network automatic modelling framework for biomedical entity and relation extraction, to resolve this challenge. In MVGNAS, we propose an automatic multi-view representation learning to learn low-dimensional representations of nodes that capture multiple relationships in a multi-view graph, representing the first research work in literature to solve the problem of multi-view graph representation learning architecture search for biomedical entity and relation extraction tasks. The experimental results demonstrate that MVGNAS can achieve the best performance in biomedical entity and relation extraction tasks against the state-of-the-art baseline methods.