Anisotropic hyperbolic polaritons (HPs) provide novel opportunities to control light at nanoscale because of their promising ability of subdiffractional confinement, low losses, and polarization sensitivities. However, it has not been explored in the modulation of many-body near-field radiative heat transfer (NFRHT). In this work, we investigate the NFRHT in an α-phase molybdenum trioxide three-body nanoparticles (NPs) system. The results show that the near-field radiative conductance (NFRHC) can be significantly enhanced by adding the third NP between two NPs due to the excitation of the localized hyperbolic polaritons (LHPs). The enhancement effect can be actively modulated by rotating the intermediate NP, which is attributed to the high directionality of the LHPs. In addition, the effects of interparticle distance and vertical displacement on the NFRHC are also studied. We achieved an enhancement ratio of over 20,000 when the interparticle distance and vertical displacement is 4 times and 1.2 times the radius, 38 times higher than that of previous works. This work demonstrates the potential of anisotropic HPs in NFRHT modulation within the many-body system.