<p>We propose a radial basis function collocation method for third-kind Volterra integral and integro-differential equations with piecewise delays. The approximation is built at scattered nodes and the integrals are evaluated by composite Gauss–Legendre quadrature on uniform subintervals. Under standard assumptions we prove stability and provide an a priori error estimate for the fully discrete scheme. The discretization reduces to a linear system for the expansion coefficients and does not require a background grid. A set of numerical examples is used to assess accuracy; the results are consistent with the analysis. A comparison with a moving least squares approach further demonstrates the robustness of the proposed method.</p>