The wiki says that Hinata has Lightning chakra nature. Granted, it's a wiki so take it with a grain of salt, but it could be that they have elemental natures that just don't get utilized for whatever reason.
everyone has an affinity... its just up to a ninja to learn how to utilize it within their ninjutsu. they don't need to spend so much time developing ninjutsu that has elemental affinity because they already have a style of fighting based on utilizing their byakugan.
I also asked myself this... Neji the genius that he is couldn't even come up with an element jutsu. He could have used it to save his life in part 1 but Kishi nerfed the plot for the sake of Uchiha pride..
They don't have a particular chakra nature....everyone's is different. They utilize their byakugan and came up with the gentle fist as their way of fighting.
Their clan doesn't have a nature transformation that they specialize in imo, or that the manga has said I think . Gentle fist is basically their speciality alongside the byakugan.