Strong gravitational lensing by galaxy clusters magnifies background galaxies, enhancing our ability to discover statistically significant samples of galaxies at z>6, in order to constrain the high-redshift galaxy luminosity functions. Here, we present the first five lens models out of the Reionization Lensing Cluster Survey (RELICS) Hubble Treasury Program, based on new HST WFC3/IR and ACS imaging of the clusters RXC J0142.9+4438, Abell 2537, Abell 2163, RXC J2211.7-0349, and ACT-CLJ0102-49151. The derived lensing magnification is essential for estimating the intrinsic properties of high-redshift galaxy candidates, and properly accounting for the survey volume. We report on new spectroscopic redshifts of multiply imaged lensed galaxies behind these clusters, which are used as constraints, and detail our strategy to reduce systematic uncertainties due to lack of spectroscopic information. In addition, we quantify the uncertainty on the lensing magnification due to statistical and systematic errors related to the lens modeling process, and find that in all but one cluster, the magnification is constrained to better than 20% in at least 80% of the field of view, including statistical and systematic uncertainties. The five clusters presented in this paper span the range of masses and redshifts of the clusters in the RELICS program. We find that they exhibit similar strong lensing efficiencies to the clusters targeted by the Hubble Frontier Fields within the WFC3/IR field of view. Outputs of the lens models are made available to the community through the Mikulski Archive for Space Telescopes