An mpox outbreak occurred suddenly and rapidly spread worldwide in 2022. Research has demonstrated a link between the sexual behavior of men who have sex with men (MSM) and the contraction of mpox. This study assessed the factors related to mpox-vaccine uptake among MSM in Taiwan, focusing on the roles of information sources and emotional problems. In total, 389 MSM participated in an online survey. Data on the participants’ vaccination statuses; anxiety symptoms, which were assessed using the State–Trait Anxiety Inventory; depressive symptoms, which were assessed using the Center for Epidemiologic Studies Depression Scale; and risk perceptions of contracting mpox were collected. Factors related to mpox-vaccine uptake were examined using a multivariable logistic regression model. The results revealed that MSM who were older (p < 0.001), perceived a higher risk of contracting mpox (p = 0.040), and received mpox information from health-care providers (p < 0.001) were more likely to receive mpox vaccination, whereas MSM who reported a greater severity of depression (p = 0.017) were less likely to receive mpox vaccination. However, age did not moderate the associations of perceiving a higher risk of contracting mpox, receiving mpox information from health-care providers, and depression with having an mpox vaccination. Health-care providers should consider these factors when developing intervention programs for enhancing mpox-vaccine uptake among MSM.