Zammad is open-source software. There are no license fees for using it, whether commercially or privately.
As long as you comply with the terms of AGPL V3, you can do anything you want with Zammad.
Will the On-Premise version remain open-source?
We can assure you that we will continue offering Zammad as On-Premise as usual. If you are worried about possible "SaaS only" or license changes, we can confirm that nothing will change. And to make sure that this is the case we have given the code to the Zammad foundation.
To show you that we not only write this FAQ post but also advertise it publicly (because we are also proud to go this way!), here is a link to the corresponding website. Feel free to have a look around!