入 means "enter", 口 means "mouth". 入口 means... "entrance". Actually for most kanji there is no single meaning. Some meanings might even have nothing in common with each other, because they've been based on ancient Chinese wordplay or something.
But 口 also means "loophole", so 入口 means "entrance" is perfectly logical - "The loophole that allows you to enter another building/strucutre".
A vomitorium (any modern person associates that with vomiting, i.e., stuff coming out of your mouth) was the name for entrances in Roman amphitheatres.