© 2017. The American Astronomical Society. All rights reserved. Many galaxy formation models predict alignments between galaxy spin and the cosmic web (i.e., directions of filaments and sheets), leading to an intrinsic alignment between galaxies that creates a systematic error in weak-lensing measurements. These effects are often predicted to be stronger at high redshifts (z 1) that are inaccessible to massive galaxy surveys on foreseeable instrumentation, but IGM tomography of the Lyα forest from closely spaced quasars and galaxies is starting to measure the z ∼ 2-3 cosmic web with requisite fidelity. Using mock surveys from hydrodynamical simulations, we examine the utility of this technique, in conjunction with coeval galaxy samples, to measure alignment between galaxies and the cosmic web at z ∼ 2.5. We show that IGM tomography surveys with ≲5 h-1Mpc sightline spacing can accurately recover the eigenvectors of the tidal tensor, which we use to define the directions of the cosmic web. For galaxy spins and shapes, we use a model parameterized by the alignment strength, , with respect to the tidal tensor eigenvectors from the underlying density field, and also consider observational effects such as errors in the galaxy position angle, inclination, and redshift. Measurements using the upcoming ∼1 deg2CLAMATO tomographic survey and 600 coeval zCOSMOS-Deep galaxies should place 3σ limits on extreme alignment models with , but much larger surveys encompassing >10,000 galaxies, such as Subaru PFS, will be required to constrain models with . These measurements will constrain models of galaxy-cosmic web alignment and test tidal torque theory at z ∼ 2, improving our understanding of the physics of intrinsic alignments.