Where do we find the Sacrament of Holy Orders in the Bible?

Summary

From a concise article The Seven Sacraments: An Introduction and Overview by a Catholic Jesuit Felix Just, S.J. which includes the biblical basis for all the 7 Catholic sacraments, these verses are what he cited for the Sacrament of Holy Orders:

• Mark 3:13-19 & par. - Jesus "calls" and "appoints" the 12 apostles
• Acts 6:6; 1 Tim 4:14; 5:22; 2 Tim 1:6 - "laying on of hands" as the "ordaining" or commissioning rite of local Christian leaders

Universal AND ministerial priesthood

The 2010 Catholic Answers article The Priesthood is Both Ministerial and Universal by Catholic apologist Tim Staples implicitly answers the Protestant argument for the priesthood of all believers.

Tim explains:

1. How Jesus is the only one mediator of the new covenant who is forever living (Heb 7:22-25) in comparison to many priests in the old covenant who necessarily died a physical death.
2. 1 Pet 2:5-9, echoing Ex 19:6 where "[Israel] shall be to me a kingdom of priests and a holy nation"), is the basis for the universal priesthood of all believers. But it does not necessarily exclude a ministerial priesthood, echoing in the same passage a separate priesthood: "... let the priests who come near to the Lord consecrate themselves..." (Ex 19:22).
3. That "priest" (Gk. hiereus) is not listed among titles of ministers such as apostles (cf. Eph 4:11), presbyters (cf. Jas 5:14), bishops (cf. 1 Tim 3:1), and deacons (cf. 1 Tim 3:8ff).
4. But hinted in the Bible is language referring to "priestly service" (Gk. hierourgounta):
   • Rom 15:15b-16a: Paul speaks of his apostolic ministry as "priestly service"
   • John 20:21-23: Jesus sending apostles to act as priests or "mediators between God and men" to forgive sin, with OT equivalent in Lv 19:21-22. Notice the present tense of "If you forgive ... they are forgiven", thus the Protestant interpretation that the sins were already forgiven is invalid.
   • James 5:13-17 implies there are ministers specifically called by God to minister to his people in his place, referring to Elijah and the rain as the OT example (1 King 17:1, 1 King 18:1).
   • Matt 16:18-19 implies how

     God’s power being released “from heaven” is contingent upon a future event of Peter and his successors acting on earth. That, in a nutshell, could be a dictionary definition of priest.

Scriptures referred to in Catholic Catechism and Lumen Gentium

For other Biblical reasoning of this sacrament (understood as apostolic succession that Christ himself instituted so that He, through the bishops, can preside over the people of God today), please read the Catholic Catechism CCC 1536 to 1600 (Vatican version here). CCC 1575-1576 is quoted below:

V. Who can confer this sacrament?

1575: Christ himself chose the apostles and gave them a share in his mission and authority. Raised to the Father's right hand, he has not forsaken his flock but he keeps it under his constant protection through the apostles, and guides it still through these same pastors who continue his work today.⁶¹ Thus, it is Christ whose gift it is that some be apostles, others pastors. He continues to act through the bishops.⁶²

1576: Since the sacrament of Holy Orders is the sacrament of the apostolic ministry, it is for the bishops as the successors of the apostles to hand on the "gift of the Spirit,"⁶³ the "apostolic line."⁶⁴ Validly ordained bishops, i.e., those who are in the line of apostolic succession, validly confer the three degrees of the sacrament of Holy Orders.⁶⁵

_{61 Cf. Roman Missal, Preface of the Apostles I.
62 Cf. LG 21; Eph 4:11.
63 LG 21 § 2.
64 LG 20.
65 Cf. DS 794 and Cf. DS 802; CIC, can. 1012; CCEO, can. 744; 747.}

Footnotes 62-64 refers to paragraphs 20 and 21 of Lumen Gentium:

20. That divine mission, entrusted by Christ to the apostles, will last until the end of the world,(147) since the Gospel they are to teach is for all time the source of all life for the Church. And for this reason the apostles, appointed as rulers in this society, took care to appoint successors.

For they not only had helpers in their ministry,(4*) but also, in order that the mission assigned to them might continue after their death, they passed on to their immediate cooperators, as it were, in the form of a testament, the duty of confirming and finishing the work begun by themselves,(5*) recommending to them that they attend to the whole flock in which the Holy Spirit placed them to shepherd the Church of God.(148) They therefore appointed such men, and gave them the order that, when they should have died, other approved men would take up their ministry.(6*) Among those various ministries which, according to tradition, were exercised in the Church from the earliest times, the chief place belongs to the office of those who, appointed to the episcopate, by a succession running from the beginning,(7*) are passers-on of the apostolic seed.(8*) Thus, as St. Irenaeus testifies, through those who were appointed bishops by the apostles, and through their successors down in our own time, the apostolic tradition is manifested (9*) and preserved.(10*)

Bishops, therefore, with their helpers, the priests and deacons, have taken up the service of the community, (11*) presiding in place of God over the flock,(12*) whose shepherds they are, as teachers for doctrine, priests for sacred worship, and ministers for governing.(13*) And just as the office granted individually to Peter, the first among the apostles, is permanent and is to be transmitted to his successors, so also the apostles' office of nurturing the Church is permanent, and is to be exercised without interruption by the sacred order of bishops. (14*) Therefore, the Sacred Council teaches that bishops by divine institution have succeeded to the place of the apostles, (15*) as shepherds of the Church, and he who hears them, hears Christ, and he who rejects them, rejects Christ and Him who sent Christ.(149)(16*)

21. In the bishops, therefore, for whom priests are assistants, Our Lord Jesus Christ, the Supreme High Priest, is present in the midst of those who believe. For sitting at the right hand of God the Father, He is not absent from the gathering of His high priests,(17*) but above all through their excellent service He is preaching the word of God to all nations, and constantly administering the sacraments of faith to those who believe, by their paternal functioning.(150) He incorporates new members in His Body by a heavenly regeneration, and finally by their wisdom and prudence He directs and guides the People of the New Testament in their pilgrimage toward eternal happiness. These pastors, chosen to shepherd the Lord's flock of the elect, are servants of Christ and stewards of the mysteries of God,(151) to whom has been assigned the bearing of witness to the Gospel of the grace of God,(152) and the ministration of the Spirit and of justice in glory.(153)

which in turn is based on these Bible verses (footnotes 147-153):

_{147 Cf. Mt. 28:20.
148 Cf. Acts 20:28.
149 Cf. Lk. 10:16.
150 Cf. 1 Cor. 4:15.
151 Cf. 1 Cor. 4:1.
152 Cf. Rom. 15:16; Acts 20:24.
153 Cf. 2 Cor. 3:8-9.}