The James Webb Space Telescope (JWST) will be instrumental in characterizing the compositions of the inner regions of protoplanetary disks. JWST will provide crucial compositional data that will complement chemical studies of the outer regions of disks probed at sub-mm/mm wavelengths and the atmospheres of gas giant planets. Several small carbon- and/or oxygen- bearing molecules can be observed at mid-infrared wavelengths including H2O, CO2, OH, HCN, C2H2, and CH4. Simulating the physical and chemical structure of a generic T Tauri disk under different initial conditions, we present models of abundances and mid-infrared emission for these molecules. The high density of photons in the surface layers of the disk result in abundances that do not reflect the initial carriers of C and O but do remain sensitive to the overall C/O ratio. We find that if we transfer gas from these photon-dominated surface layers to the midplane, this gas will not return to an interstellar-like composition rich in carbon-bearing ices on timescales of less than a few million years. The degree to which surface gas compositions sampled via mid-infrared observations mirror those of the planet-forming regions in the disk midplane will depend on the efficiency of vertical mixing. Modeled molecular fluxes are sensitive to the C/O ratio, gas temperature, and inner gas radius of the disk. High gas temperatures of up to 800 K result in CH4 fluxes that are comparable to those of the other observable molecules, whereas moderate temperatures, less than 400 K, result in much lower CH4 fluxes. Therefore, observations of CH4 in protoplanetary disks may provide evidence for heating of inner disk gas due to processes beyond stellar irradiation.