In autonomous driving, the high-definition (HD) map plays a crucial role in localization and planning. Recently, several methods have facilitated end-to-end online map construction in DETR-like frameworks. However, little attention has been paid to the potential capabilities of exploring the query mechanism for map elements. This paper introduces MapQR, an end-to-end method with an emphasis on enhancing query capabilities for constructing online vectorized maps. To probe desirable information efficiently, MapQR utilizes a novel query design, called scatter-and-gather query, which is modelled by separate content and position parts explicitly. The base map instance queries are scattered to different reference points and added with positional embeddings to probe information from BEV features. Then these scatted queries are gathered back to enhance information within each map instance. Together with a simple and effective improvement of a BEV encoder, the proposed MapQR achieves the best mean average precision (mAP) and maintains good efficiency on both nuScenes and Argoverse~2. In addition, integrating our query design into other models can boost their performance significantly. The source codes will be released online.