Background/Objectives: Understanding the factors that influence tendon mechanical properties is essential for optimizing performance and preventing injuries in elite athletes. This study aimed to identify the strongest correlates of the biomechanical properties (frequency, stiffness, logarithmic decrement, relaxation, and creep) in the Achilles and patellar tendons in elite international athletes. Methods: A cross-sectional study was conducted with 111 elite athletes from 11 sports disciplines assessed at a high-performance training center. Tendon properties were measured bilaterally using MyotonPRO. Anthropometric (height, weight, age), demographic (sex, limb dominance defined as the preferred limb for sport-specific activities), and sport-specific variables were analyzed using correlation, multiple regression, and machine learning approaches. Results: Height showed the strongest correlations with tendon frequency and stiffness, particularly for the Achilles tendon (r = 0.52 for frequency; r = 0.53 for stiffness; p ≤ 0.001, large effects). Sex differences were evident across all measures, with men showing higher stiffness and frequency, and women greater relaxation and creep (partial η² = 0.35–0.48, Cohen’s d = 0.84–1.16). Sports discipline explained substantial variance in tendon properties (η² > 0.40), and limb dominance influenced Achilles stiffness, with left-dominant athletes showing higher values (p < 0.05). Age showed minimal associations (r < 0.10). Conclusions: Height, sex, and sports discipline were the strongest correlates of Achilles and patellar tendon mechanical properties in elite athletes, with large and practically meaningful effects across sports. This comprehensive analysis, utilizing multivariate and machine learning approaches, provides insights that can inform individualized training, injury prevention, and performance optimization strategies in high-performance sports.