"I invite you!"
Ehm... if the situation makes it obvious, to doing what 'other stuff' you're inviting...
It's too weird, it would only seem okay if you put accent on JA to show that it's you who is inviting, and not your twin brother. "Zapraszam cię/was" in this order and without the very often redundant pronoun seems like the only natural version.
If i just said 'zapraszam' to someone, if we had already been talking about the thing to which I was inviting them, wouldn't I be saying 'I invite you'?
Oh, yes, we added it some time after your comment. But only if the context makes it really clear what you mean.
From the English sentence, you have no idea. Therefore you just choose one, both should be not only accepted but considered 'best answer'.
„Zapraszam cię/ciebie na kawę”. - correct version. We often invite someone somewhere („do” „do domu, do kina, do restauracji, do mnie”) or to some activity/thing („na”, „na kawę”, „na koncert”, „na wycieczkę”). These two rules have exceptions, so if one wants to dance with someone he says: „zapraszam do tańca” or sth like that. We also have „zapraszam nad morze, nad jezioro, w góry ”
„Zapraszam ciebie dla kawę?” - „dla” usually denotes the benefactor one for whom the action is being made
Frankly, I'd rather simply go with the literal translation of "come in" and say "Wchodźcie"/"Wejdźcie" (one is imperfective, the other is perfective, don't ask me why both work, I have no clue).
"Zapraszam" without any pronoun would make sense to me for "come in", but with a pronoun, that sounds strange. "Witajcie" is "Welcome", which in a way works.
(Wchodź/Wejdź/Witaj for singular 'you')