After clicking the 'Sign in with Apple' button on the Expo ID login screen, if you close the 'Sign in with Apple' pop-up without clicking 'Continue', you will be unable to click the 'Sign in with Apple' button again.
If this occurs, refresh the browser page on the Expo ID login screen to enable the 'Sign in with Apple' button, then click it again and click 'Continue' on the 'Sign in with Apple' pop-up.