With wavelength tunability, free-electron lasers (FELs) are well-suited for generating orbital angular momentum (OAM) beams in a wide photon energy range. We report here the first experimental demonstration of OAM beam generation using an oscillator FEL with the tens of picosecond pulse duration. Lasing around 458 nm, we have produced the four lowest orders of superposed Laguerre-Gaussian beams using a very long FEL resonator of 53.73 m. The produced beams have good beam quality, excellent stability, and substantial average power. We have also developed a pulsed operation mode for these beams with a highly reproducible temporal structure for a range of repetition rate of 1-30 Hz. This development can be extended to short wavelengths, for example to x-rays using a future x-ray FEL oscillator. The OAM operation of such a storage-ring FEL also paves the way for the generation of OAM gamma-ray beams via inverse Compton scattering.