Yes, you can drive on almost any road in New Zealand, no matter the type of road. However, we do recommend taking our Stress-Free Insurance when driving a lot on gravel roads or unsealed roads as then any damages caused by these rough roads is covered.
When damages to the campervan happen when driving on gravel roads, it'll be the same process as for other damages, your excess will be used to pay for the repair the vehicle.
See for an overview of the insurance options. When you have Stress-Free Insurance damages like that will be covered by the insurance and there's no bond/excess.
Also, have a look at this overview of the various types of roads in New Zealand and their quality. That will give you a better understanding of what kind of roads to expect in New Zealand.
There are only 3 roads in New Zealand you are not allowed to drive, these are the 3 restricted roads:
Skippers Canyon Road near Queenstown
Ball Hutt Road near Mt Cook
90 Mile Beach in Northland
Also, you must not - unless authorised by us - drive or take the campervan on any road, which is temporarily/permanently closed or restricted, or on beaches through streams, dams, rivers and floodwaters.