/ AbstractSteady gamma-ray emission up to at least 200 GeV has been detected from the solar disk in the Fermi-LAT data, with the brightest, hardest emission occurring during solar minimum. The likely cause is hadronic cosmic rays undergoing collisions in the Sun's atmosphere after being redirected from ingoing to outgoing in magnetic fields, though the exact mechanism is not understood. An important new test of the gamma-ray production mechanism will follow from observations at higher energies. Only the High Altitude Water Cherenkov (HAWC) Observatory has the required sensitivity to effectively probe the Sun in the TeV range. Using three years of HAWC data from November 2014 to December 2017, just prior to the solar minimum, we search for 1--100 TeV gamma rays from the solar disk. No evidence of a signal is observed, and we set strong upper limits on the flux at a few $10^{-12}$ TeV$^{-1}$ cm$^{-2}$ s$^{-1}$ at 1 TeV. Our limit, which is the most constraining result on TeV gamma rays from the Sun, is $\sim10\%$ of the theoretical maximum flux (based on a model where all incoming cosmic rays produce outgoing photons), which in turn is comparable to the Fermi-LAT data near 100 GeV. The prospects for a first TeV detection of the Sun by HAWC are especially high during solar minimum, which began in early 2018.