Q: Does a person need to be ordained in order to baptize another believer?
A: Concerning the question of who is qualified to baptize, nothing in the Bible requires the baptizer to be an ordained minister, or an elder, or even a deacon in a local church. Because the act of baptizing is a spiritual function, it is something I believe the Bible limits to males. But any Christian male in a right relationship with the Lord is qualified to baptize.
I do make one minor recommendation on this subject. Because the ritual of water baptism has all too often been connected to a specific denomination or as the means to joining a specific church, I always suggest that baptizing be done in a neutral place — such as a beach, a lake, river, perhaps even a private swimming pool. This may help separate the ritual from other non-related issues such as church membership, etc. Water baptism is an outward ritual of one's inward faith, just as communion is an outward ritual or representation of the spiritual truth of that which Messiah accomplished through His sacrificial death.
Water baptism is not the means by which new converts are welcomed into the Body of Yeshua, as it is Spirit baptism that actually places one into the Body.