The development of control methods in vehicle platoon with irregular constraints has emerged as a research topic of considerable interest. This article investigates the adaptive tracking control for vehicle platoon systems with irregular constraints and input saturation. First, a sufficiently differentiable shift function is designed to distinguish clearly between constrained and unconstrained states by adding transition processes so that the range for the constrained state can be preset and smooth transitions between two states are guaranteed. Subsequently, an enhanced nonlinear extended state observer is developed to estimate external disturbances by proposing a new Gaussian-based Faln function, which eliminates the nondifferentiability at switching points that could compromise platoon safety. Moreover, input saturation and unknown parameters are tackled through the Nussbaum-type function and adaptive backstepping framework, respectively. Using Lyapunov stability theory, it is proved that all signals of the platoon system are semiglobally uniformly ultimately bounded. Comparative simulations on a vehicle platoon system demonstrate the effectiveness of the proposed scheme.