We present the first multiepoch spectropolarimetry of a Type II plateau supernova (SN II-P), with optical observations of SN 1999em on days 7, 40, 49, 159, and 163 after discovery. These data are used to probe the geometry of the electron scattering atmosphere before, during, and after the plateau phase, which ended roughly 90 days after discovery. Weak continuum polarization with an unchanging polarization angle (θ ≈ 160^°) is detected at all epochs, with p ≈ 0.2% on day 7, p ≈ 0.3% on days 40 and 49, and p ≈ 0.5% in the final observations. Distinct polarization modulations across strong line features are present on days 40, 49, 159, and 163. Uncorrected for interstellar polarization (which is believed to be quite small), polarization peaks are associated with strong P Cygni absorption troughs, and nearly complete depolarization is seen across the Hα emission profile. The temporal evolution of the continuum polarization and sharp changes across lines indicate polarization intrinsic to SN 1999em. When modeled in terms of the oblate, electron scattering atmospheres of Höflich, the observed polarization suggests an asphericity of at least 7% during the period studied. The temporal polarization increase may indicate greater asphericity deeper into the ejecta. We discuss the implications of asphericity on the use of Type II-P supernovae as primary extragalactic distance indicators through the expanding photosphere method (EPM). If asphericity produces directionally dependent flux and peculiar galaxy motions are characterized by σ = 300 km s^-1, it is shown that the agreement between previous EPM measurements of SNe II and distances to the host galaxies predicted by a linear Hubble law restricts mean SN II asphericity to values less than 30% (3 σ) during the photospheric phase.