Biometric verification systems are deployed in various security-based access-control applications that require user-friendly and reliable person verification. Among the different biometric characteristics, fingervein biometrics have been extensively studied owing to their reliable verification performance. Furthermore, fingervein patterns reside inside the skin and are not visible outside; therefore, they possess inherent resistance to presentation attacks and degradation due to external factors. In this paper, we introduce a novel fingervein verification technique using a convolutional multihead attention network called VeinAtnNet. The proposed VeinAtnNet is designed to achieve light weight with a smaller number of learnable parameters while extracting discriminant information from both normal and enhanced fingervein images. The proposed VeinAtnNet was trained on the newly constructed fingervein dataset with 300 unique fingervein patterns that were captured in multiple sessions to obtain 92 samples per unique fingervein. Extensive experiments were performed on the newly collected dataset FV-300 and the publicly available FV-USM and FV-PolyU fingervein dataset. The performance of the proposed method was compared with five state-of-the-art fingervein verification systems, indicating the efficacy of the proposed VeinAtnNet.