We present SMA and NOEMA observations of the host galaxy of FRB 121102 in the CO 3-2 and 1-0 transitions, respectively. We do not detect emission from either transition. We set $3sigma$ upper limits to the CO luminosity $L_{CO} < 2.5 times 10^7,{rm K,km,s}^{-1} {, rm pc^{-2}}$ for CO 3-2 and $L_{CO} < 2.3 times 10^9, {rm K,km,s}^{-1} {, rm pc^{-2}}$ for CO 1-0. For Milky-Way-like star formation properties, we set a $3sigma$ upper limit on the $H_2$ mass of $2.5 times 10^8 rm M_{odot}$, slightly less than the predictions for the $H_2$ mass based on the star formation rate. The true constraint on the $H_2$ mass may be significantly higher, however, because of the reduction in CO luminosity that is common forlow-metallicity dwarf galaxies like the FRB host galaxy. These results demonstrate the challenge of identifying the nature of FRB progenitors through study of the host galaxy molecular gas. We also place a limit of 42 $mu$Jy ($3sigma$) on the continuum flux density of the persistent radio source at 97 GHz, consistent with a power-law extrapolation of the low frequency spectrum, which may arise from an AGN or other nonthermal source.
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