Naturally, as this recipe is from Japan, it is traditionally made with sushi rice, which is a staple in most Japanese rice dishes. However, since the rice in this dish simply makes up the base of the dish and isn't required to add structure (as it would in sushi) then you can swap it for any rice variety of your choosing.
Sushi rice has a short grain and is stickier than most other rice varieties as it contains more starch. The texture will be quite different. Therefore, it's really up to you if you prefer the stickiness of sushi rice with the rich teriyaki sauce.
If you want to avoid sushi rice because you don't like the flavour, then likely this is because of the addition to rice vinegar which is typically used to season sushi rice. Therefore you could skip the vinegar rather than changing the rice. Conversely, if you can only find long grain rice and want to make it taste like sushi rice then you can simply add the vinegar.