Reactive agility (RA) directly refers to athletes’ visuomotor processing of the specific conditions for team sports. The aim of the study was to identify the factors among age, gender, sport discipline, time participation in a sports activity, reaction time, and visual field which could have an impact on visual-motor performance in RA tasks in young, competitive team sports players. The study included boys (n = 149) and girls (n = 157) aged 13–15 participating in basketball, volleyball and handball. Anthropometric measurements were carried out, and the Peripheral Perception (PP) test was used to evaluate the visual-motor performance under laboratory conditions. The Five-Time Shuttle Run to Gates test was used to determine the RA. A multiple regression analysis was performed to identify the relationships between the visual-motor performance in an RA task (dependent variable) and the remaining independent variables (continuous and categorical). The findings of the current study indicate that the main predictive factors of visual-motor performance in RA among young athletes are gender (ß = −0.46, p < 0.000) and age (ß = −0.30, p < 0.000). Moreover, peripheral perception positively affected the achievements in the RA task in boys (ß = −0.25, p = 0.020). The sport discipline does not differentiate the visual-motor performance in RA in team sports players in the puberty period.