It is not the first time I have seen 'enter into' which is used in such a way. Do you think if it is correct or not?
Taking Refuge Vows is the gateway through which we enter into Buddhism and become a Buddhist.
I disagree with Raymott in the context. 'To enter into' here means to become involved/engaged in ('they entered into a relationship'/they entered into an agreement'). You would never 'enter Buddhism'; you might 'enter into Buddhism'; more likely, you would 'embrace Buddhism'.