Prep the Ingredients: Bring the rice, water and vinegar to a boil in a saucepan over high heat. Reduce the heat to medium-low then cover while simmering until the rice is tender and the liquid has been absorbed; 20 to 25 minutes. Let the rice stand covered for about 10 minutes more to absorb any excess water. Set rice aside to cool.
Lightly mix together the tuna, mayonnaise, chilli and wasabi in a bowl being sure to break the tuna apart but not mashing it into a paste.
To Roll the Sushi: Cover a bamboo sushi rolling mat with plastic wrap. Lay a sheet of nori, rough side up, on the plastic wrap. With wet fingers, firmly pat a thick, even layer of the prepared rice over the nori to cover it completely.
Place about 1 tablespoon each of cucumber, carrot and avocado in a line along the bottom edge of the sheet then spread a line of tuna mixture alongside the vegetables.
Pick up the edge of the bamboo rolling sheet then fold the bottom edge of the sheet up to enclosing the filling. Tightly roll the sushi into a thick cylinder. Once the sushi is rolled wrap it in the mat and gently squeeze to compact it tightly.
Cut each roll into 6 pieces and refrigerate until served.