We present a cosmological analysis of the Lick Observatory Supernova Search (LOSS) Type Ia supernova (SN Ia) photometry sample introduced by Ganeshalingam et al. (2010ApJS..190..418G, Cat. J/ApJS/190/418). These supernovae (SNe) provide an effective anchor point to estimate cosmological parameters when combined with data sets at higher redshift. The data presented by Ganeshalingam et al. have been rereduced in the natural system of the Katzman Automatic Imaging Telescope (KAIT) and Nickel telescopes to minimize systematic uncertainties. We have run the light-curve-fitting software SALT2 on our natural-system light curves to measure light-curve parameters for LOSS light curves and available SN Ia data sets in the literature. We present a Hubble diagram of 586 SNe in the redshift range z=0.01-1.4 with a residual scatter of 0.176mag. Of the 226 low-z SNe Ia in our sample, 91 objects are from LOSS, including 45 without previously published distances. Assuming a flat Universe, we find that the best fit for the dark energy equation-of-state parameter w=-0.86^+0.13^_-0.16_(stat)+/-0.11(sys) from SNe alone, consistent with a cosmological constant. Our data prefer a Universe with an accelerating rate of expansion with 99.999% confidence. When looking at Hubble residuals as a function of host-galaxy morphology, we do not see evidence for a significant trend, although we find a somewhat reduced scatter in Hubble residuals from SNe residing within a projected distance <10kpc of the host-galaxy nucleus ({sigma}=0.156mag). Similar to the results of Blondin, Mandel and Kirshner and Silverman et al. (2012AJ....143..126B, Cat. J/AJ/143/126), we find that Hubble residuals do not correlate with the expansion velocity of SiII{lambda}6355 measured in optical spectra near maximum light. Our data are consistent with no presence of a local "Hubble bubble". Improvements in cosmological analyses within low-z samples can be achieved by better constraining calibration uncertainties in the zero-points of photometric systems.