The dynamic interaction between moving vehicles and bridge structures represents one of the most complex coupled problems in structural engineering. This paper presents a comprehensive coupled finite element method (FEM) and multibody system (MBS) framework for analysing vehicle–bridge interaction (VBI) dynamics. The bridge is discretised using Euler–Bernoulli beam elements, while the vehicle is modelled as a multi-degree-of-freedom (MDOF) sprung mass system. Road surface roughness is generated stochastically using the ISO 8608 power spectral density (PSD) function, and the coupled equations of motion are integrated numerically using the Newmark-β time-stepping algorithm. The dynamic amplification factor (DAF), mid-span deflection, and internal stress responses are evaluated across a parametric range of vehicle speeds (20–140 km/h), axle loads (80–320 kN), and roughness classes (A–C). Results demonstrate that DAF increases nonlinearly with speed and roughness severity, reaching values up to 1.68 for Class C surfaces at 140 km/h, exceeding many current design code limits. A global sensitivity analysis using Sobol indices identifies vehicle speed and axle load as the dominant parameters governing bridge response. The study provides actionable guidance for bridge design codes and maintenance prioritisation, particularly in developing-nation infrastructure contexts.