The principal purpose of this investigation is to present an approximate algorithm for solving third-kind Volterra integral equation with nonlinear vanishing delays using mesh-free techniques based on a radial basis function collocation scheme. This technique does not require background approximation cells, and the algorithm is efficient, has good stability and does not require much computer memory. A description of the approach for the proposed equations is provided. The error estimation of the proposed approach is also provided. The accuracy and reliability of the new scheme are demonstrated through numerical examples. Some of these examples have been compared with analytical solutions and moving least squares methods. The numerical results validate their agreement with theoretical error estimates.