ANSWER: There is nothing in the Bible that states who can and cannot baptize someone. If such a simple ceremony was forbidden to be carried out except by those who serve God fulltime, then one would be hard pressed to explain one of the greatest events that occurred on the day of Pentecost! When we take a closer look at the church's first celebration of Pentecost, we find it would have been IMPOSSIBLE for a small group of men (the apostles) to have been able to baptize several thousand people in a single day!
Peter, on Pentecost, started speaking at about 9:00 AM (Acts 2:15). When he finished, God caused about three thousand people to repent of their sins and want to be baptized (Acts 2:41). Assuming his sermon took about an hour leaves us with about eight hours of daylight left in the day (480 minutes). Let us also assume that the twelve apostles were the only ones to baptize the crowd. This would mean each of them, of necessity, would have to perform this ceremony on 250 people. This means that each apostle would only have less than TWO MINUTES to baptize a repentant believer IF they worked non-stop without breaks! Clearly, this would be virtually impossible.
Consider, however, if all the one hundred and twenty disciples (Acts 1:15) present on Pentecost and who received God's spirit played their part to baptize 3,000 people. Each one would only have to perform this ceremony on only 25 people within the same period of eight hours! In this scenario, the person doing the baptism could use up to fifteen minutes of their time on each person and STILL have plenty of time for breaks and to eat.
Again, the Bible does NOT state or teach that a person who has a particular title, rank or status in the church MUST be the only person to baptize new believers in the faith. It was God, after all, who commanded Ananias (an "average" church member who was not someone we would label as "ordained"), to perform a pivotal act that would resound through Christian history. He was told to baptize a young, repentant traveler from Jerusalem named Saul (who was later renamed Paul), a man who would ultimately become the second most influential Christian in history next to Jesus (Acts 9:10 – 11, 17 - 18)!
The apostle Paul did not believe that the ability to perform a baptism was some special 'right' bestowed only on those who were considered fulltime ministers. 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 (1Corinthians 1:14 - 17).
There is no Biblical prohibition that states church members cannot baptize new Christians 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.