A couple of weeks ago I've bought this Carmona bonsai. It grows beautifully, plus already has some new light green leaves. It is potted in organic bonsai soil.

As you can see on the photos, the roots are exposed in certain areas, it bothers me and I don't think it's good for the plant.
Submerge watering makes things worse, so I avoid that method.

Can you give me some advice to fix this? How should I fill the whole pot and keep the soil together? Is repotting necessary?

A few weeks ago I repotted a dawn redwood forest that has been in the same kind of Coco mix for two years after purshase. It was a similar situation. Lots of hollows in the soil/rootball. However not the optimal for developing a nice root structure it did not pose any harm to the health of the trees.

I stand firm in my belief that beginners are best to grow trees a year or two. Learning the species and its care before repotting.

I would recoment to add a drainage screen to the pot and fill the hollows with a non organic substrate.