There is nothing in Scripture that says only ordained clergy can administer communion. Since we are all priests in the kingdom of God (1 Pet. 2:5, 9), anyone who is a Christian can administer communion. However, it is wise to have someone who is knowledgeable about communion be the one who oversees it. Essentially, since communion is such an important event that represents the sacrifice of Christ, and because it is an official ceremony in the church, it seems proper that officials such as elders and pastors be the ones who administer this sacred ceremony.
Also, we would not want someone who is in blatant sin to administrate; nor would we want anyone who has no understanding of the nature of God, the nature of the sacrifice, etc., to offer Communion and maybe do it in a casual and flippant manner.
Since communion was instituted by Christ in an official capacity, this is probably why the custom of having ordained clergy, who are officials in the church, be the only ones who administer communion.