Thank you for the question on Melchizedek, “Who ordains Melchizedek as a priest in the Bible?”
The Biblical text is silent on the “ordination” of Melchizedek. Thus it would be conjecture, but here are some possibilities.
A. It is possible that the king of Salem appointed himself to be priest over his people. He was the king-priest.
Genesis 14 says this, “18 Then Melchizedek king of Salem brought out bread and wine. He was priest of God Most High,” (NIV). This one king-priest recognized and worship God Most High.
B. It is possible that the king of Salem inherited the title of priest over his people. It may have been passed onto him from his forefathers who ruled over the city of Salem.
C. It is possible that God revealed to the king of Salem that he is the One and True God. For that revelation, the people recognized him as priest since God revealed himself to Melchizedek.
D. It is possible that Melchizedek came to a deeper realization of the God of Abram when he heard how God rescued the people and possessions from a mightier army. It wasn’t because of Abram’s wisdom or might that brought about the victory.
Genesis 14 says this, “and he blessed Abram, saying, “Blessed be Abram by God Most High, Creator of heaven and earth. 20 And praise be to God Most High, who delivered your enemies into your hand.” (NIV).
From reading the text, here’s my thought. Melchizedek had the title of king of Salem and priest over Salem. This was an inherited position. The encounter with Abram enabled him to realize that his position of king and priest came by God for God delivered him from the hands of the four kings. He acknowledged that the God of Abram is greater than all other gods.
SUMMARY: It is likely that Melchizedek inherited the title of King of Salem and Priest of Salem.