When users configure email or form mails to work with cPanel in theory if scripted properly messages from forms should be delivered as well as any email for the domain.
However in some cases users prefer to use 3rd party MX records with the domain indicating that the mail is setup with another party such as Google or Yahoo for example. In this case there is no issue with using third party email providers, however you must ensure that your domain is set to look for remote MX records. If you are on a Virtual Dedicated or a Dedicated Server then this can be configured in the settings of the platform, however if you are using a Shared hosting account you may need to contact your hosting provider in order to have them change your domains to look for remote MX records.