Being a musician usually entails playing an instrument, but there are downsides to having a drum set or a six-string, especially when lugging one's own equipment on tour.
Sigor Rós: This Icelandic band sings all their songs in a non-literal "language" called Vonleska. Vonleska has no grammar or syntax, and focuses solely on the sound of language. Emotive howls, fast-paced gibberish, and melodic strings of syllables comprise the sounds, but there are no inherent concepts or meanings behind the tones (unlike the words and sentences in real languages).
Bjork: Another import from Iceland, Bjork is known more for singing in intelligible English than using her voice as an instrument, but her album Medulla is almost entirely a cappella, and the songs are made mostly from layers of Bjork's vocals. There are few instruments on the record; throat singing, beatboxing, and choral arrangements dominate. And the album features an appearance from another vocal instrumentalist on this list, Rahzel.
Bobby McFerrin: Anyone who could hear in the 1980s remembers McFerrin's "Don't Worry, Be Happy" all over the radio. Although most of McFerrin's songs incorporate his ability to layer vocal harmonies into symphonic-sounding melodies, "Don't Worry, Be Happy" is the best and most well-known example of McFerrin's skills. The track is rich with sounds and whistling melodies, but there's not an inorganic instrument to be found here. The sounds of violin strings, softly thumping drums, and steady bass guitar are all made from multi-track layers of McFerrin's voice.