What vegetables should I add to a japanese curry?
The most common ones simmered along with curry are onions, carrots, and potatoes. You could even add some sliced apples. Root vegetables are frequently simmered with the curry, and you could consider using variants like sweet potato or squash, kabu, daikon, etc.

As a "topping", the sky is the limit; I've seen blanched okra, cooked renkon, roasted or simmered slices of kabocha squash, green beans, sweet potatoes, and cauliflower.

As non-vegetable toppings added after simmering, I've seen cheese, katsu-style tofu, tonkatsu or menchi katsu.

Edited to add: as Japanese curry is one of the easiest ways of encouraging our children to eat vegetables, my wife and I now both add anything from blanched spinach (near the end of cooking) to (typically canned) tomatoes; the younger one loves it when there are plenty of green peas or even corn in the mix. I like to add cauliflower.