We will be implementing IPV6 in addition to our IPV4 addresses.
The question is, is there a way to apply an ipv4 and an ipv6 address to the same website in HCnix?

From Hosting Business point of view its not ideal to use both IPv4 and IPv6 for a single website. So, better not to use multiple ips against one website. If there is some specific requirement for your webapplication then can you share it here.

We are integrating ipv6 addressing over the next few months to eliminate the need to use reassigned IPV4 blocks from our internet connection provider.
It was our intention to try to run dual ips for websites until IPV6 is fully implemented.

You can change your website IP from IPv4 to IPv6 ipaddress in HC panel by navigating to this menu : Hosting Services :: Manage Websites :: My Websites :: Website Properties :: Change IP.
 
So, you can make your websites to run with IPv4 address and once you plan to move them to different ipaddress, you can always make this change from within the control panel from the aforementioned path.