It seems to think "Vacation will begin in June" is wrong (the system accepted it as a typo, assuming I'd meant "vacations"), but I find it a natural thing to say in US English. One would say this especially in the sense of "summer vacation" (i.e. when school is closed).
Given how similar Present Tense and Future Simple may be in Polish, we are very strict in terms of using tenses in the translation. "zaczną się" is Future Simple, therefore we require "will begin/are going to begin". Your sentence is "zaczynają się".