The Fermilab Muon $g-2$ experiment aims to measure the anomalous magnetic moment of the muon to a precision of 140 ppb, a fourfold improvement over the previous E821 measurement at BNL. The measurement is extracted from an analysis of the modulation of the rate of higher-energy positrons from muon decays, detected by 24 calorimeters and 3 straw tracking detectors. The straw tracking detectors will be used to measure the beam profile, cross-calibrate the calorimeter, identify pile-up and track muons lost from the storage region. In addition, a tracker measurement of the vertical modulation of positrons will be used for a search for an electric dipole moment signal. Herein, the readout electronics of the straw tracker are described. Analogue signals from the straws are amplified, shaped and discriminated by custom ASDQ ASICs and the digital output is passed to an FPGA-based TDC which records the hit time with a resolution of 0.6 ns. The hits are buffered and event records are created by FC7 and AMC13 modules in a MicroTCA crate, which are then passed into the MIDAS event-builder and data acquisition system.