Uh no. Not only does the establishment clause NOT prohibit the government promoting religion, bit it EXCLUSIVELY applies to the government.
So a person not affiliated with the school, outside school grounds (or even inside school grounds) can hand out bibles all they want. Even a teacher could hand out bibles AS LONG AS the teacher doesn't require the pupils to become a member of a specific religion. The teacher CAN hand out bibles as a teaching aid in comparative religion for example, he just can't mandate that the children go to church and pay tithes.
That's a major flaw in many peoples' understanding of the establishment clause (mainly people on the left, but also some on the right), it does NOT prohibit religious expression, it only prohibits MANDATING OTHERS from joining a religion, AKA establishing a state mandated religion. This was put into the constitution to prevent a situation arising like it had in Britain where the government outlawed all religions except the Church of England and made membership of that religion mandatory for everyone.
ORIGINALLY it didn't even apply to the individual states, only to the federal government, and several states did indeed have a state religion!
So no, handing out bibles doesn't violate the first amendment. In fact prohibiting that person from handing out bibles WOULD violate the first amendment as it'd restrict their freedom to express their opinion, thus their freedom of speech.