We have used Chandra observations of eight bulgeless disk galaxies to identify new ultraluminous X-ray source (ULX) candidates, study their high-mass X-ray binary (HMXB) population, and search for low-luminosity active galactic nuclei (AGNs). We report the discovery of 16 new ULX candidates in our sample of galaxies. Eight of these are found in the star forming galaxy NGC 337, none of which are expected to be background contaminants. The HMXB luminosity function of NGC 337 implies a star formation rate (SFR) of 6.8_-3.5_^+4.4^M_{sun}/yr, consistent at 1.5{sigma} with a recent state of the art SFR determination. We also report the discovery of a bright ULX candidate (X-1) in ESO 501-23. X-1's spectrum is well fit by an absorbed power law with {Gamma}=1.18_-0.11_^+0.19^ and N_H_=1.13_-1.13_^+7.07^/cm2, implying a 0.3-8keV flux of 1.08_-0.07_^+0.05^x10^-12^erg/s/cm2. Its X-ray luminosity (L_X_) is poorly constrained due to uncertainties in the host galaxy's distance, but we argue that its spectrum implies L_X_>10^40^erg/s. An optical counterpart to this object may be present in an Hubble Space Telescope image. We also identify ULX candidates in IC 1291, PGC 3853, NGC 5964, and NGC 2805. We find no evidence of nuclear activity in the galaxies in our sample, placing a flux upper limit of 4x10^-15^erg/s/cm2 on putative AGN. Additionally, the Type II-P supernova SN 2011DQ in NGC 337, which exploded two months before our X-ray observation, is undetected.