  Methanol (CH$_3$OH) is at the root of organic ice chemistry in protoplanetary
disks. However, its weak emission has made detections difficult. To date,
gas-phase CH$_3$OH has been detected in only one Class II disk, TW Hya. We use
the Atacama Large Millimeter/submillimeter Array (ALMA) to search for a total
of four CH$_3$OH emission lines in bands 6 and 7 toward the disk around the
young Herbig Ae star HD 163296. The disk-averaged column density of methanol
and its related species formaldehyde (H$_2$CO) are estimated assuming optically
thin emission in local thermodynamic equilibrium. We compare these results to
the gas-phase column densities of the TW Hya disk. No targeted methanol lines
were detected individually nor after line stacking. The 3$\sigma$
disk-integrated intensity upper limits are $< 51$ mJy km s$^{-1}$ for the band
6 lines and $< 26$ mJy km s$^{-1}$ for the band 7 lines. The band 7 lines
provide the strictest 3$\sigma$ upper limit on disk-averaged column density
with $N_{\mathrm{avg}} < 5.0 \times 10^{11}$ cm$^{-2}$. The
methanol-to-formaldehyde ratio is CH$_3$OH/H$_2$CO $< 0.24$ in the HD 163296
disk compared to a ratio of $1.27$ in the TW Hya disk. Differences in the
stellar irradiation of Herbig disks compared to T Tauri disks likely influence
the gaseous methanol and formaldehyde content. Possible reasons for the lower
HD 163296 methanol-to-formaldehyde ratio include: a higher than expected
gas-phase formation of H$_2$CO in the HD 163296 disk, uncertainties in the
grain surface formation efficiency of CH$_3$OH and H$_2$CO, and differences in
the disk structure and/or CH$_3$OH and H$_2$CO desorption processes that
release the molecules from ice mantles back into the gas phase. These results
provide observational evidence that the gas-phase chemical complexity found in
disks may be strongly influenced by the spectral type of the host star.
