Who Can Baptize, Administrators

Below are some excerpts from confessions of faith in which the compilers address the topic of who had the authority to administer baptism -- sometimes specifically, as in ordained ministers and sometimes more generally, as under the authority of the church. There are many associations and conventions whose articles of faith do not address the subject of the administrator of the ordinances at all. The Baptist Faith and Message 2000 of the Southern Baptist Convention does not, but I have included it here because (1) they are the largest worldwide group of Baptists, and (2) some believe calling it a "church ordinance" implies church authority in administration. The list skews conservative, partly because more conservative churches feel a greater need to address such a topic. The list is presented first chronologically in the order the confession was composed or last revised, and second alphabetically for those confessions for which finding a date was more complicated than I wanted to deal with at the moment.

Chronological 

Dordrecht Anabaptist/Mennonite Confession of Faith, 1632 
That the apostles, likewise, as faithful followers of Christ, and leaders of the church, were diligent in this respect, with prayer and supplication to God, through the election of brethren, to provide every city, place, or church, with bishops, pastors, and leaders, and to ordain such persons thereto, who would take heed unto themselves, and unto the doctrine and flock, who were sound in faith, pious in life and conversation, and of good report without as well as in the church; that they might be an example, light, and pattern in all godliness and good works, worthily administering the Lord's ordinances--baptism and supper-and that they might everywhere (where such could be found) appoint faithful men who would be able to teach others also, as elders, ordaining them by the laying on of hands in the name of the Lord, and provide for all the wants of the church according to their ability...

London Baptist Confession of 1644 
The persons designed by Christ, to dispense this ordinance [of baptism], the Scriptures hold forth to a preaching Disciple, it being no where tied to a particular church, officer, or person extraordinarily sent, the commission enjoining the administration, being given to them under no other consideration, but as considered Disciples. 

London Baptist Confession of Faith, 1646 
The person designed by Christ to dispense baptism, the Scripture holds forth to be a disciple; it being no where tied to a particular church officer, or person extraordinarily sent the commission enjoining the administration, being given to them as considered disciples, being men able to preach the gospel. 

Appendix to the London Confession of Faith 1646, by Benjamin Cox 
A disciple gifted and enabled by the Spirit of Christ to preach the Gospel, and stirred up to this service by the same Spirit, bringing home to his soul the command of Christ in His word for the doing of this work, is a man authorized and sent by Christ to preach the Gospel...And such preachers of the Gospel may not only lawfully administer baptism unto believers, and guide the action of the church in the use of the Supper...

Second London Baptist Confession of Faith, 1689 (Also Philadelphia Confession) 
26.8: A particular church, gathered and completely organized according to the mind of Christ, consists of officers and members; and the officers appointed by Christ to be chosen and set apart by the church (so called and gathered), for the peculiar administration of ordinances, and execution of power or duty, which he intrusts them with, or calls them to, to be continued to the end of the world, are bishops or elders, and deacons. (Acts 20:17, 28; Philippians 1:1)
28.2: These holy appointments are to be administered only by those who are qualified and thereunto called, according to the commission of Christ. (Matthew 28:19; 1 Corinthians 4:1)

Kehukee Association Articles of Faith (North Carolina), 1777 
We believe that no minister has no right to administration of the ordinances, only as are regularly called and come under the imposition of hands by the presbytery. 

Primitive Baptist Confession of Faith, Fulton, Kentucky, 1900 
...administering baptism by immersion to penitent believers only by ministers of the gospel clothed with authority by the Gospel Church... 

Treatise of the Faith and Practices of the Free Will Baptists, 1948 
Duties of Ministers. These are to preach the Word, administer the ordinances of the Gospel, visit their people, and otherwise perform the work of faithful ministers. 

The Baptist Faith and Message, 2000 
Christian baptism is the immersion of a believer in water in the name of the Father, the Son, and the Holy Spirit...Being a church ordinance, it is prerequisite to the privileges of church membership and to the Lord's Supper. 

Articles of Faith of Baptist Bible Fellowship International, 2002 
We believe that Christian baptism is the immersion in water of a believer, in the name of the Father, of the Son, and of the Holy Ghost, with the authority of the local church... 

Articles of Faith of Global Independent Baptist Fellowship, 2002 We believe baptism is the immersion in water of a believer in the name of the Father, of the Son, and of the Holy Ghost, by and under the authority of a New Testament Baptist church... 

Alphabetical 

Articles of Faith of the National Primitive Baptist Convention, USA, Inc.
We Believe that Baptism is the immersion of a believer in water by a proper administrator (an ordained elder) in the name of the Father, Son, and the Holy Ghost. 

Articles of Faith of the Sardis Association of Old Regular Baptist
We believe that no Minister has the right to administer the ordinances and commands of the Gospel except such as are regularly ordained and baptized, and that by immersion by a legal administrator of the Gospel come under the hands of a regularly chosen presbytery of the Church.

Doctrinal Statement of the American Baptist Association 
Scriptural baptism is the immersion of penitent believers in water, administered by the authority of a New Testament church in the name of the Father, Son, and Holy Spirit. 

Doctrinal Statement of the Baptist Missionary Association of America 
The Ordinances of the Church--Her two ordinances are baptism and the Lord's Supper...Both ordinances must be administered by the authority of a New Testament church.

Statement of Principles of the Interstate and Foreign Landmark Missionary Baptist Association
...the church, as set up in Jerusalem by the Lord Jesus Christ, and her successors, are the only institutions authorized to administer these Ordinances...

Notes: 

• The Kehukee Association (org. 1769) is about the fifth oldest association of Baptist churches in America. Its statement (or slight variations of it) on the administrator of the ordinances was widely disseminated in association articles of faith throughout the South. 
• A church or association's statement of administering the ordinances falling under the duties of ministers may not necessarily mean that they strictly limit the administration of ordinances to ministers.