Website and Roundcube Webmail Transfer

[Barriez]

The email for my small business is Roundcube Webmail hosted by a website company.  I am transferring our website to a different company that does not host emails.  How will I keep these roundcube emails open, active and functioning?  How do I transfer the emails so that we do not lose them?  The new website hosted by a different company is almost ready but they advised me if I do not have a place to transfer these emails we will lose them all.  Please help or can roundcube support please contact me?  Thank you!