Terahertz vortices prompt numerous advanced applications spanning classical and quantum communications, sensing, and chirality-based detection, owing to the inherent physical properties of terahertz waves and orbital angular momentum (OAM). Nonetheless, existing methodologies for generating terahertz vortices face challenges such as unalterable topological charges and intricate feed networks. To address these limitations, we propose a novel approach to generate multi-mode and tunable vortex beams based on chiral plasmons. Through eigenmode analysis, the uniform helical gratings are demonstrated to support chiral plasmons carrying OAM. By leveraging their vortex characteristics and introducing modulation into the periodic system, these chiral plasmons are alternatively diffracted into high-purity vortex radiations according to the Bragg law. To validate the theory, the vortex beam emitter is fabricated and measured in the microwave regime based on the modulated scheme. Experimental results confirm the emission of vortex beams with desirable phase distributions and radiation patterns. Our findings highlight the potential of chiral plasmons as seeds for tunable and compact vortex radiation, offering promising applications in tunable vortex sources.