Is it possible to use Javascript to emulate the Add to Home Screen option in Mobile Safari's bookmark menu?
Something similar to IE's window.external.AddFavorite(location.href, document.title); possibly?
TLDR: @Craig has a better answer above.
Below is my original answer, but I do not believe it answers the question adequately. Today I would say you'd have to try a library to emulate this effect, as A2HS is not supported in WebViews (PWAs).
@Kerrick I would like to delete my answer, but cannot as it has been accepted.
Old answer:
Yes. The majority of modern browsers support the Add to Home screen (or A2HS) feature for Progressive Web Apps. To quote the Mozilla Web Docs article:
Add to Home screen is a feature available in modern browsers that allows a user to "install" a web app, ie. add a shortcut to their Home screen.
See also: A2HS browser support at caniuse.com