The Muon g-2 experiment at Fermilab will measure the anomalous magnetic moment of the muon, $a^{}_\mu$ with a precision of 140ppb, a four-fold improvement over the former BNL E821 result. If the current discrepancy of about 3.5$\sigma$ between the Standard Model value for $a^{}_\mu$ and the former experiments was real, then the new experiment would verify the deviation with >7$\sigma$ significance. The experiment uses muons with the magic momentum of 3.09GeV/c stored in a very homogeneous magnetic field of a 45m long storage ring. The determination of $a^{}_\mu$ requires the precise knowledge of both the muon spin precession, $\omega^{}_a$, and the magnetic field, $\omega^{}_p$, expressed in terms of the free-proton Larmor frequency. The field measurement uses the high precision tool of pulsed proton Nuclear Magnetic Resonance and requires cross-calibrating a set of probes: a very accurate NMR probe calibrates the 17 probes on the in-vacuum field mapping system (the so-called trolley). This trolley in turn is used to map the field over the full azimuth of the storage ring. The trolley measurement also provides the calibration of the 378 fixed probes around the ring that constantly monitor the field drift. The trolley is particularly important as it measures the relevant field that is seen by the stored muons and this article presents the details of this field mapping system. This work was supported in part by the US DOE, Fermilab and US DOE OHEP under contract No. KA2201020.