The issue is very simple - google.com is blocked in some countires (such as China), so reCaptcha won't work there. The usual workaround is to replace google.com with recaptcha.net.
How would I do this in wpForo? I understand wpForo itself has a built-in integration with reCaptcha, but its settings page doesn't include any relevant options for this. I suppose I need to modify some wpForo file?
Just if anyone's curious: I've already set up reCaptcha for my wpForo forum and it works perfectly well; it just doesn't work for users visiting from Chinese network.
I don't think you can do what you ask with wpForo forms.
So you can use the standard Wordpres login/register pages and i guess in there you can use whatever you want.