We report measurements of quantum turbulence generated by a vibrating grid in superfluid $^3$He-B at zero pressure in the zero temperature limit. Superfluid flow around individual vortex lines Andreev-reflects incoming thermal ballistic quasiparticle excitations, and allows non-invasive detection of quantum vortices in $^3$He-B. We have compared two Andreev reflection-based techniques traditionally used to detect quantum turbulence in the ballistic regime: quasiparticle transmission through and reflection from ballistic vortex rings and a turbulent tangle. We have shown that the two methods are in very good agreement and thus complement each other. Our measurements reveal that vortex rings and a tangle generated by a vibrating grid have a much larger spatial extent than previously realised. Furthermore, we find that a vortex tangle can either pass through an obstacle made from a mesh or diffuse around it. The measured dependence of vortex signal as a function of the distance from the vibrating grid is consistent with a power-law behaviour in contrast to turbulence generated by a vibrating wire which is described by an exponential function.