Well, I really have no idea why it fails @William, some guys tried the api setup it worked well and some other users of phpvms v7 are using mailgun with api too. I am not blaming you, there should be something blocking api access somewhere. Classic smtp settings should work, I mean it is basic mailing, works for every server, why it should fail at yours.
You already have server, port, username, password etc for their smtp access. Just use the default SMTP example with that info.
SMTP
MAIL_MAILER=smtp
MAIL_FROM_NAME='American Virtual'
MAIL_FROM_ADDRESS='postmaster@americanva.org'
MAIL_HOST=smtp.mailgun.org
MAIL_PORT=587
MAIL_ENCRYPTION=
MAIL_USERNAME='postmaster@americanva.org'
MAIL_PASSWORD='that long password here'
API
MAIL_MAILER=mailgun
MAILGUN_DOMAIN='americanva.org'
MAILGUN_SECRET='that long api key here'
I assume you are cleaning the application cache from admin > maintenance after doing changes to env.php and saving/uploading that file, failing or forgetting this step will cause problems.
(technically until you clean the cache your old env.php will remain in effect and changes will not be applied)