Howdy! Thank you for reporting the issue. I finished consulting with other devs who experienced the same issue in other projects and here's the full troubleshooting suggestions.
Under certain circumstances Orc Breeding may fail to run. This occurs because the engine we use (Twine) requires Web Storage to be completely functional. Private browsing modes (In Private, Incognito, the Brave browser) break Web Storage, among other things, in the name of enhanced privacy. Thus, Orc Breeding generally won't function completely or at all in such environments.
If none of these options work, disabling 3rd-party cookies can also do it, since that setting also controls Web Storage. Using Itch.io's desktop app should also resolve the issue.
Sorry for taking a while!