We theoretically propose and experimentally generate the nondiffracting Bessel-Poincaré beams whose Stokes vortices radially accelerate during propagation. To this end, we design the Bessel beams whose intensity is specified to be uniformly distributed along the longitudinal direction. By superposing two such Bessel beams having different helical phases and mutually orthogonal polarizations, the synthesized vector beam is endowed with the polarization singularity that can rotate about the optical axis, while the total intensities maintain their profiles. Radially self-accelerating Stokes vortices in the resulting beam can be manipulated by adjusting the predefined parameters in the constituent beams.