An asymptotic normalization constant (ANC) is a measure of the probability in the large-radius tail of a wave function. It is probed in some laboratory experiments where atomic nuclei exchange particles, and it can be used to predict the cross sections of various astrophysical nuclear reactions. In particular, alpha-particle capture by a 3H or 3He nucleus to form a 7Li and 7Be is of great interest for Big Bang nucleosynthesis and energy generation in hydrogen-burning stars, and those capture rates are directly related to ANCs of the final-state nuclei. In this work, I carry out ab initio calculations of ANCs for alpha-particle removal in 7Li and 7Be states. I implement the calculations as extensions of the standard nuclear variational Monte Carlo (VMC) code. The ANCs are computed as integrals over the accurately-computed interior of the wave function, applying a method that has not been used previously in ab initio calculations of alpha-removal ANCs. The new integral method should be insensitive to inaccuracies in the computed wave function tails that are difficult to avoid. I give some simple demonstrations of the method for toy models, present a thorough derivation of its application to many-body wave functions, and explain its implementation in the VMC code. After presenting ANCs for the bound 7Li and 7Be systems, I explore their sensitivity to the wave function variational ansatz, and I examine the consistency of the computed ANCs with the short-range properties of the wave functions and with experimental data. Finally, I apply a direct extension of the ANC method to estimate alpha-emission widths of several unbound resonances, finding fair agreement with experimental data in favorable cases. This work opens the way toward calculations of many alpha-emission ANCs and widths in light nuclei, as well as extensions to alpha scattering cross sections and application of the integral method with more-exact wave functions computed from the Green’s function Monte Carlo method.