In this paper, we reformulate the fourth-order differential equation with boundary condition (FBVP) as a Volterra-Fredholm integral equation (VFIE). We establish the existence, uniqueness, and convergence of the solution within the VFIE framework using Picard’s method, Banach’s fixed-point theorem, and Schwarz’s inequality. This study investigates the Tau method’s performance when employing Legendre basis functions for solving FBVP. We develop an algebraic equivalent formulation of FBVP using the operational Tau matrix approach. Numerical experiments using the Tau method on linear and nonlinear FBVP demonstrate an exponential decrease in errors. Illustrative examples are given to verify the accuracy and performance of the proposed method. The numerical results include comparisons with existing results for the same problems, which clearly highlight the accuracy of the proposed Tau method.