It is a long-standing goal of nuclear physics to develop accurate first-principles models of the structures and reaction properties of nuclei. Such models are important for astrophysics because of the difficulties often involved in direct measurements of reactions. However, nuclear reactions present very difficult cases of the quantum many-body problem. A mitigating circumstance is the domination of some astrophysical photon-emission reactions by the asymptotic regions of the nucleus at large radius. This allows calculation of approximate rates in simple two-body models, where some input parameters are determined from many-body calculations of the final-state nuclei. We present calculations of the asymptotic normalization coefficients (ANCs) of light-nucleus wave functions from accurate variational Monte Carlo many-body wave functions. The ANCs are computed from integrals over the wave function interior that are related to the Lippmann-Schwinger equation. This method is in fact more accurate than direct examination of the wave function surface region, due to errors in the wave function and technical difficulties in the examination. We focus in particular on ANCs for alpha-particle removal from 7Li and 7Be, which are related to important reaction rates for solar neutrinos and the Big Bang.