ANSWER: There is nothing in the Bible that states who can and cannot perform the simple but profound ceremony known as baptism.
Let us take a brief look at the most well-known mass baptism in the New Testament - the one which occurred on the day of Pentecost. If we take a close enough look we will find that it would have been nearly IMPOSSIBLE for a group of eleven men (the apostles) to have taken care of the baptismal needs God generated that day!
On Pentecost, Peter started speaking at about 9:00 AM (Acts 2:15). When he finished, God caused about three thousand people to repent of their sins, accept Jesus as their Savior, and request to be baptized (Acts 2:41). If we assume Peter's sermon took about an hour, that would leave only about eight hours (480 minutes) left of daylight. Let us also assume that only the twelve apostles did all the baptisms and 'laying on hands' on the believer in order to pray God gives him his Spirit. This would mean each apostle, of necessity, would have to baptize 250 people (3,000 divided by 12). They would only have less than TWO MINUTES to do what is required for each repentant believer IF they worked non-stop for eight continuous hours without any breaks! In short, it would be nearly impossible for twelve people to do all the work.
Consider, however, if all the one hundred and twenty disciples (Acts 1:15) present on Pentecost and who received God's spirit participated in baptizing new believers. Each one would have to baptize about 25 people in eight hours - a far more realistic approach. In this scenario, the person doing the baptizing could use up to fifteen minutes of their time on each person and STILL have plenty of time for breaks and to eat.
Each baptism would, according to Scripture, be done "in the name of Jesus" (Acts 4:8 - 12, Colossians 3:17).
The Bible does NOT state nor teach that a person who has a particular title, rank or status in the church MUST perform a baptism or lay hands on the new believers in order for them to receive God's spirit. God commanded Ananias, whom Scripture does not refer to as an "ordained" minister, evangelist, church elder, apostle, etc. but was rather referred to simply as "a certain disciple," to baptize Saul (who was later renamed Paul).
10. Now there was in Damascus a certain disciple named Ananias. And the Lord said to him in a vision, "Ananias." And he said, "Behold, I am here, Lord." 11. And the Lord said to him, "Arise and go into the street which is called Straight, and inquire in the house of Judas for one named Saul from Tarsus; for behold, he is praying . . .
17. Then Ananias went away and came into the house; and after laying his hands on him, he said, "Brother Saul, the Lord has sent me, even Jesus, Who appeared to you on the road in which you came, so that you might receive sight and be filled with the Holy Spirit." 18. And it was as if scales immediately fell from his eyes, and he instantly received sight; and he arose and was baptized (Acts 9:10 – 11, 17 - 18, HBFV).
The apostle Paul did not believe that the ability to baptize someone was some special 'right' bestowed on those who were considered church elders or "ordained" into the ministry. It was not meant to be something only a select few could perform or some special responsibility requiring the church to make sure only those who were "approved" perform the simply ceremony. Paul, in fact, references the ability to perform this service as something he personally preferred not to do (and rarely did unless circumstances required it) and which he did not track. In correcting the Corinthian church for their foolish arguments over who is the "greatest" Biblical teacher he states the following.
14. I thank God that I did not baptize any of you, except Crispus and Gaius, 15. Lest someone should say that I baptized new converts into my own name. 16. But I also baptized the household of Stephanas; as for any others, I do not know if I baptized anyone else. 17. For Christ did not send me to baptize, but to preach the gospel - not with the wisdom of words, lest the cross of Christ be made void (see 1Corinthians 1:14 - 17, HBFV)
There is no Biblical prohibition that states church members cannot perform a baptism or that such an act would be invalid in God's eyes. Those who state members should never be allowed to perform such a simple service are likely relying far more on their church's traditions and policies than the sacred words of Scripture.