For basic functionality (new user, lost password, comment moderation) your WordPress instance needs to be able to send email.
=== CiviCRM ===
If you integrate [[CiviCRM]] into your WordPress installation, then you might also have configured CiviCRM independently to hand off email to MailGun or another provider through the SMTP configuration. Using MailGun is preferred (over plain 'mail()' or even GSuite) when your WordPress website is being used to send monthly messages (even just a few thousand) to your newsletter and other list subscribers.
=== CLI ===
WordPress has a command line interface that lets you administer the system; even remotely. See also [[CiviCRM/WordPress]]
For example, you can quickly get a list of your users, with ID, login, display_name, user_email, user_registered and roles