Gamma-ray bursts (GRBs) have long been considered a possible source of high-energy neutrinos. While no correlations have yet been detected between high-energy neutrinos and GRBs, the recent observation of GRB 221009A-the brightest GRB observed by Fermi-GBM to date and the first one to be observed above an energy of 10 TeV-provides a unique opportunity to test for hadronic emission. In this paper, we leverage the wide energy range of the IceCube Neutrino Observatory to search for neutrinos from GRB 221009A. We find no significant deviation from background expectation across event samples ranging from MeV to PeV energies, placing stringent upper limits on the neutrino emission from this source.