Basically, you don’t need anything. However, you should bring favorite toppings, even though soy sauce and ponzu (soy sauce with citrus juice) are already available in the huts.
Most of the huts allow customers to bring in their own toppings. Many people bring in butter, mayonnaise, Chinese chili bean paste, Kimchi, olive oil or pizza sauce and such.
The good thing is that you can try in many ways as you like, share them within your groups.
In my case, I brought in butter, olive oil, cheese, baguette and grated radish with chili powder.