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We present measurements of the redshift-dependent clustering of a DESI-like luminous red galaxy (LRG) sample selected from the Legacy Survey imaging dataset, and use the halo occupation distribution (HOD) framework to fit the clustering signal. The photometric LRG sample in this study contains 2.7 million objects over the redshift range of $0.4 < z < 0.9$ over 5655 deg$^2$. We have developed new photometric redshift (photo-$z$) estimates using the Legacy Survey DECam and WISE photometry, with $σ_{\mathrm{NMAD}} = 0.02$ precision for LRGs. We compute the projected correlation function using new methods that maximize signal-to-noise ratio while incorporating redshift uncertainties. We present a novel algorithm for dividing irregular survey geometries into equal-area patches for jackknife resampling. For a five-parameter HOD model fit using the MultiDark halo catalog, we find that there is little evolution in HOD parameters except at the highest redshifts. The inferred large-scale structure bias is largely consistent with constant clustering amplitude over time. In an appendix, we explore limitations of Markov chain Monte Carlo fitting using stochastic likelihood estimates resulting from applying HOD methods to N-body catalogs, and present a new technique for finding best-fit parameters in this situation. Accompanying this paper we have released the Photometric Redshifts for the Legacy Surveys (PRLS) catalog of photo-$z$'s obtained by applying the methods used in this work to the full Legacy Survey Data Release 8 dataset. This catalog provides accurate photometric redshifts for objects with $z < 21$ over more than 16,000 deg$^2$ of sky.