Optioncart is an inventory database manager that I’m considering. It requires a CPanel hosting account. I assume that’s what I have here, but can anyone verify that for me?

Google is your friend on this, but generally nothing really needs a CPanel hosting account. that is just a method by which you access your hosting resources. In their own site… they specify the requirements:

You need MySQL & PHP and a hosting account of course. All of them available with DH offerings.

Hope this helps.