Detailed chemical abundance distributions for 14 elements are derived for eight high-probability stellar members of the solar metallicity old open cluster M67 with an age of similar to 4 Gyr. The eight stars consist of four pairs, with each pair occupying a distinct phase of stellar evolution: two G dwarfs, two turnoff stars, two G subgiants, and two red clump (RC) K giants. The abundance analysis uses near-IR high-resolution spectra (lambda 1.5-1.7 mu m) from the Apache Point Observatory Galactic Evolution Experiment survey and derives abundances for C, N, O, Na, Mg, Al, Si, K, Ca, Ti, V, Cr, Mn, and Fe. Our derived stellar parameters and metallicity for 2M08510076+1153115 suggest that this star is a solar twin, exhibiting abundance differences relative to the Sun of <= 0.04 dex for all elements. Chemical homogeneity is found within each class of stars (similar to 0.02 dex), while significant abundance variations (similar to 0.05-0.20 dex) are found across the different evolutionary phases; the turnoff stars typically have the lowest abundances, while the RCs tend to have the largest. Non-LTE corrections to the LTE-derived abundances are unlikely to explain the differences. A detailed comparison of the derived Fe, Mg, Si, and Ca abundances with recently published surface abundances from stellar models that include chemical diffusion provides a good match between the observed and predicted abundances as a function of stellar mass. Such agreement would indicate the detection of chemical diffusion processes in the stellar members of M67.