In this paper, a complete set of real-valued entire-domain vector basis functions is developed on a circular domain. The basis functions are constructed based on the orthogonal Zernike polynomials, and they satisfy the boundary conditions of vanishing normal components at the rim of the domain. This study discovers a number of new mathematical relations of the radial Zernike polynomials, which allows the method of moments (MoM) impedance matrix to be obtained in a completely analytical manner without the need of meshing or numerical integrations. An empirical, sufficient condition for the convergence of an MoM solution with the proposed basis is presented. As an example, the developed theories are applied to the determination of the characteristic modes of circular conducting plates, and their effectiveness and efficiency are demonstrated by comparing to results in the literature and those obtained by using the RWG sub-domain basis functions. The validity of the empirical convergence condition is also verified.