The stable propagation of orbital angular momentum and cylindrical vector beams in a newly designed annular core photonic crystal fiber (AC-PCF) tailored for the broadband single-radial order beam transmission (within the so-called “endlessly mono-radial” guiding regime) is demonstrated for the first time. It is shown that the vector-vortex beams can maintain high mode purities above 18 dB after propagation in the fiber under test over all of the wavelength range from 805 to 845 nm (over 17 THz bandwidth) investigated with the help of a tunable laser and an S-plate for the generation of singular beams in free space. Our results confirm that the AC-PCF is a promising design for the broadband transmission of vector-vortex beams that have potential applications in space-division multiplexing, quantum communications, optical sensing and trapping.