The primary goal of the Carnegie Chicago Hubble Program (CCHP) is to measure the Hubble Constant by using Population II standard candles to calibrate the Type Ia supernova (SN Ia) distance scale, the most precise and accurate probe of the Hubble Flow available. So far, the CCHP has measured direct distances to 11 SNe Ia, and here we increase that number to 15 with two new TRGB distances measured to NGC 5643 and NGC 1404, for a total of 20 SN Ia calibrators. We present resolved, point-source photometry from new Hubble Space Telescope (HST) imaging of these two galaxies in the F814W and F606W bandpasses. In each galaxy's F814W-band luminosity function we detect an unambiguous edge feature which is identified as the Tip of the Red Giant Branch (TRGB). To measure the true distance to each galaxy, we combine the observed TRGB magnitudes with a calibration of the absolute TRGB magnitude that has been verified using two independent geometric "anchor" galaxies: (1) the Large Magellanic Cloud (LMC), with a distance determined from detached eclipsing binaries (DEBs) and (2) NGC 4258, with a distance determined via the orbital motion of maser clouds located near its central black hole. As a result, we determine the first set of distances to these two galaxies measured with a primary distance indicator. For NGC 5643, we find \(\mu_0 = 30.48\pm0.03(stat)\pm0.07(sys) \) mag, and for NGC 1404 we find \( \mu_0=31.36\pm 0.04(stat)\pm 0.05(sys)\) mag. From a preliminary consideration of the SNe Ia in these galaxies, we find increased confidence in the results presented in \citet{freedman_2019}. The precision of our TRGB distance allows us to measure between the Fornax Cluster galaxies NGC 1404 and NGC 1316 a three-dimensional displacement \(D_{1404} - D_{1316} = -1.50^{+0.25}_{-0.39}\) Mpc, which we show is in agreement with independent literature constraints.