We investigate spatially-resolved emission-line ratios in a sample of 219 galaxies (0.6 < z < 1.3) detected using the G102 grism on the Hubble Space Telescope Wide Field Camera 3, taken as part of the CANDELS Lyα Emission at Reionization (CLEAR) survey, to measure ionization profiles and search for low-luminosity active galactic nuclei (AGN). We analyze [O III] and Hβ emission-line maps, enabling us to spatially resolve the [O III]/Hβ emission-line ratio across the galaxies in the sample. We compare the [O III]/Hβ ratio in galaxy centers and outer annular regions to measure ionization gradients and investigate the potential of sources with nuclear ionization to host AGN. We investigate some of the individual galaxies that are candidates to host strong nuclear ionization and find that they often have low stellar mass and are undetected in X-rays, as expected for low-luminosity AGN in low-mass galaxies. We do not find evidence for a significant population of off-nuclear AGN or other clumps of off-nuclear ionization. We model the observed distribution of [O III]/Hβ gradients and find that most galaxies are consistent with small or zero gradients, but 6-16% of galaxies in the sample are likely to host nuclear [O III]/Hβ that is ∼0.5 dex higher than in their outer regions. This study is limited by large uncertainties in most of the measured [O III]/Hβ spatial profiles, therefore deeper data, e.g, from deeper HST/WFC3 programs or from JWST/NIRISS, are needed to more reliably measure the spatially resolved emission-line conditions of individual high-redshift galaxies.