I've camped a lot in Denmark, less so in Sweden.
The direct ferries to Scandinavia stopped a few years back. There is a possibility that Regina line may start in time for this summer but I wouldn't bank on it. Otherwise you could get a ferry to Holland (from Harwich or Hull) and drive up. Driving in Denmark and Sweden is easier than Netherlands- far less traffic apart from the main cities (but even then it's not that bad).
Or, if you fly then there are many Danish sites which offer a 'hut' to save you taking your own gear. I have never hired one but they always looked popular when spotted on sites. Some are much like camping pods- literally a shed to sleep in but others have a mini kitchenette/bathroom. One main difference with Danish campsites is that most have a kitchen for all campers anyway, with hotplates/gas, microwaves etc. You still have to take pots/plates/cutlery but it does save on other cooking equipment (though a little stove/kettle to make drinks with by your tent is often nice too).
Danish school holidays are mid June to Early August. If you were thinking of going then, then you would need to book, and you might find that you are tied to a week in each hut. After that, the sites are less full and much easier to find a hut/pitch with more flexibility.
My fav places are Skagen, Copenhagen, Faaborg, Aarhus/Ebeltoft and Bornholm. It's great fun getting various bridges or ferries around Denmark to save on driving too. There's legoland if that might float your boat and a tonne of viking historical places and castles- there's so much history as well as things to see and do.
Let me know if you want any other advice or information.