Gravitational waves are indeed everywhere. If the early universe underwent inflation, they should be everywhere. Any object orbiting another object emits gravitational waves. These exotic events, however, are the only gravitational waves that are observable using the terrestrial light interference technique (LIGO and Virgo). A space based mission would be sensitive in a different frequency range (satellites bouncing light off one another over millions of kilometers). The primordial gravitational waves from inflation might be observed indirectly -- there was much excitement about the BICEP telescope observing indirect evidence of these gravitational waves, but I believe that other explanations have come forth.