phpinfo() returns this particular setting which is problematic:
mail.log /var/log/phpmail.log
The problem is I can\'t access the phpmail
I would first try to contact whoever is managing your server, he or she should fix this problem.
If that doesn't work, the mail.log setting is a PHP_INI_PERDIR setting, meaning that:
Entry can be set in php.ini, .htaccess, httpd.conf or .user.ini (since PHP 5.3)
Obviously, you can't access the system's php.ini & httpd.conf, but if a .user.ini is provided to you, or you're allowed to change settings in .htaccess, then that would be a way to solve this.
See:
.user.ini files
Set php.ini Values Using .htaccess
Alternatively, you could consider using Swift Mailer or PHPMailer, which don't need the mail() function. In general, using mail() is discouraged, it's very basic and fairly difficult to use correctly & securely.