Background: Internships are a mandatory graduation requirement to help medical students transition to the work environment. Some individuals are prone to anxiety in an unfamiliar environment, which is a public concern among young adults. Here, we investigated the mechanism between internet gaming disorder and anxiety and insomnia among internship students. Methods: A convenient sample of 267 internship students was collected in a cross-sectional study module between 17 July and 27 December 2022. The survey contained a 7-item Generalized Anxiety Disorder (GAD-7), Athens Insomnia Scale (AIS), and Internet Gaming Disorder Scale—Short-Form (IGDS9-SF). The association was estimated using Pearson’s correlations, and network analysis was performed to characterize these associations. Results: Our results indicate that about 60% of participants exhibited mild to severe anxiety and insomnia, while 2.28% showed symptoms of internet gaming disorder. Also, we found a moderate association between anxiety and insomnia. An item-level analysis indicated that GAD_1 “feeling anxious” and GAD_5 “unable to sit still” are essential for gaming, and that GAD_2 “uncontrollable worrying” is crucial for insomnia. This indicated an interplay between these items, supported by our centrality analysis, where we found that GAD_1 and GAD_2 depicted high centrality. Conclusions: We found high rates of anxiety and insomnia in internship students and the association between selected symptoms of anxiety and insomnia. At the same time, low rates of internet gaming disorder could be attributed to a lack of time for entertainment and an increased awareness of its risks. Given these findings, an awareness of anxiety and insomnia risk should be emphasized.