Edit for the third time:
So on MAL, Tsuyuno's name is written like this "Tsuyuno Amami" but we know her sister's name is "Suzuno", so "Tsuyuno" isn't the surname, "Amami" is. Also, I asked my Japanese friend regarding how her surname is read, and he said that he hasn't seen a surname like that, but if he were to guess how to read it, he'd go with "Amamizu" or "Usui". I'm leaning more toward "Amamizu Tsuyuno" but of course that can/might change depending on later chapters.
TL;DR: Tsuyuno's full name as of now is "Amamizu Tsuyuno".