WordPress plugins will use wp_mail function to post email notification.
This wp_mail function will need PHP mail function to work.
It will work if your web hosting company allows PHP mail function.
In this case, your web hosting must have disallowed it, or maybe they have a specific set of requirements for web form.
If they only allows SMTP mail, you will have to get the setting details from them.
Use this plugin, you will be able to reconfigure the wp_mail function to use SMTP instead of PHP mail
By the way, Please check that you mails are not caught as spam on mail server or desktop mail client, before proceeding to use the above plugin.