There is nothing in the Bible that states who can and cannot baptize someone. If such a simple ceremony could not be performed except by those who are “ordained” then one would be hard pressed to explain what took place on Pentecost in 30 A.D.!
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 (e.g. the apostles) 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 request someone baptize them (Acts 2:41). Assuming his sermon took about an hour leaves us with about eight hours of daylight left in the day (480 minutes).
A real life example
If the twelve apostles were the only ones baptizing on Pentecost this would mean each of them would have to perform a baptism on 250 people. Performing such a monumental task in only eight hours, without any breaks, 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 offering to baptize could use up to fifteen minutes of their time on each person and still be able to take breaks and 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), 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 (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).
In conclusion, there is no Biblical prohibition against non-ordained but otherwise mature church members carrying out the responsibility to baptize people. 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.