To answer the OP, it depends on where you go to church.
If you are Catholic (or Orthodox, though the language is not identical), it would not do you any good for a "regular" believer to administer the Eucharist.
1365 Because it is the memorial of Christ's Passover, the Eucharist is also a sacrifice. Catechism of the Catholic Church
As a sacrifice, the Catholic Church believes only a duly ordained priest acting as Christ may offer.
1548 In the ecclesial service of the ordained minister, it is Christ himself who is present to his Church as Head of his Body, Shepherd of his flock, high priest of the redemptive sacrifice, Teacher of Truth. This is what the Church means by saying that the priest, by virtue of the sacrament of Holy Orders, acts in persona Christi Capitis: CCC 1548
It is the same priest, Christ Jesus, whose sacred person his minister truly represents. Now the minister, by reason of the sacerdotal consecration which he has received, is truly made like to the high priest and possesses the authority to act in the power and place of the person of Christ himself (virtute ac persona ipsius Christi).24 Christ is the source of all priesthood: the priest of the old law was a figure of Christ, and the priest of the new law acts in the person of Christ
So, the compelling reason for not offering as a lay person is it won't do you any good.
On the other hand, Protestants view the Thanksgiving as a memorial and never as a (re)sacrifice.
What does the Bible say (emphasis mine)?
For I have received of the Lord that which also I delivered unto you, That the Lord Jesus the same night in which he was betrayed took bread: 1 Cor 11:23
Who is the "you" to whom Paul is addressing?
Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you. verse 2
Brethren. Fellow believers. Followers of Christ, as Paul was.
It is far beyond the scope of the answer to go into this, but God reinstated the priesthood of believers in Christ.
So, fear not, the Lord is with y'all.