Hebrews 7 and 8 - Who is the New Covenant with? 2018

Heb 8:8
For finding fault with them, He says, "BEHOLD, DAYS ARE COMING, SAYS THE LORD, WHEN I WILL EFFECT A NEW COVENANT WITH THE HOUSE OF ISRAEL AND WITH THE HOUSE OF JUDAH;

The New Covenant as described here is a direct quote from Jeremiah 31:31-34.
Jeremiah 31:31-34 “Behold, days are coming,” declares the LORD, “when I will make a new covenant with the house of Israel and with the house of Judah, 32not like the covenant which I made with their fathers in the day I took them by the hand to bring them out of the land of Egypt, My covenant which they broke, although I was a husband to them,” declares the LORD. 33“But this is the covenant which I will make with the house of Israel after those days,” declares the LORD, “I will put My law within them, and on their heart I will write it; and I will be their God, and they shall be My people. 34“And they shall not teach again, each man his neighbor and each man his brother, saying, ‘Know the LORD,’ for they shall all know Me, from the least of them to the greatest of them,” declares the LORD, “for I will forgive their iniquity, and their sin I will remember no more.”

Who is the New Covenant with? It is not with "the New Testament Church" or any new group or organization. It is made with "Israel" and "Judah." We are a part of Israel (and hence a part of the New Covenant) if we have a heart for His Torah.

Eph. 2:12-13 remember that you were at that time separate from Christ, excluded from the commonwealth of Israel, and strangers to the covenants of promise, having no hope and without God in the world. 13But now in Christ Jesus you who formerly were far off have been brought near by the blood of Christ.

Eph 2:19 So then you are no longer strangers and aliens, but you are fellow citizens with the saints, and are of God’s household,

Eph 3:6 to be specific, that the Gentiles are fellow heirs and fellow members of the body, and fellow partakers of the promise in Christ Jesus through the gospel,

Rom. 11:25 For I do not want you, brethren, to be uninformed of this mystery, lest you be wise in your own estimation, that a partial hardening has happened to Israel until the fulness of the Gentiles has come in;
Paul starts this section with a discussion of Melchizedek. in order to properly set up chapter 7, we need to take a running start and look at the end of the last chapter (Heb. 6:19-20), He was a priest of the Father before the Levites were even conceived. Paul writes that Messiah is the anchor of our soul and our high priest forever according to theorder of Melchizedek.

Melchizedek comes onto the scene early in Genesis after Abram saves his nephew, Lot, from Chedarloamer (Ked-ar-la-omar) and three other kings. He is greeted by the king of Sodom and Melchizedek. He is discussed in Gen. 14:18-20.