Hi, Mooky.
Actually, Hizikyah's response was headed in the right direction, but he didn't take you all the way home, so to speak. IOW, Hizikyah was seeking to show you where what you cited from Romans 12:19 was originally "written" so that you might better understand the context of the same. Here, then, are Romans 12:19 and Deuteronomy 32:35-36:
"Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord." (Romans 12:19)
"To me belongeth vengeance, and recompence; their foot shall slide in due time: for the day of their calamity is at hand, and the things that shall come upon them make haste. For the LORD shall judge his people, and repent himself for his servants, when he seeth that their power is gone, and there is none shut up, or left." (Deuteronomy 32:35-36)
Whenever we see such phrases as "for it is written" or "according to the scriptures" in our Bibles, it would behoove us to go back and find out exactly where "it is written" so that we can best understand exactly what is being said or the reason behind the "proof text" which has been given to prove the author's intended point. You may have noticed that I not only cited Deuteronomy 32:35 where this actual quote was originally found, but that I also included the next verse where we read "for the LORD shall judge HIS PEOPLE". I did the same for a reason. IOW, BOTH of these verses are quoted in another place in the New Testament and by reading the context of the same and then coupling it with the context of where the verses originated, I believe that you will find the answer to your question. In the book of Hebrews, we read:
Hebrews chapter 10
[26] For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins,
[27] But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries.
[28] He that despised Moses' law died without mercy under two or three witnesses:
[29] Of how much sorer punishment, suppose ye, shall he be thought worthy, who hath trodden under foot the Son of God, and hath counted the blood of the covenant, wherewith he was sanctified, an unholy thing, and hath done despite unto the Spirit of grace?
[30] For we know him that hath said, Vengeance belongeth unto me, I will recompense, saith the Lord. And again, The Lord shall judge his people.
[31] It is a fearful thing to fall into the hands of the living God.
Here, the author's intent was to show that if Christians "sin willfully after having received the knowledge of the truth, there remains no more sacrifice for their sins, BUT A CERTAIN FEARFUL LOOKING FOR OF JUDGMENT AND FIERY INDIGNATION WHICH SHALL DEVOUR GOD'S ADVERSARIES". Remember, the author was addressing HEBREW CHRISTIANS or those who would have been familiar with the God of the Old Testament. As such, he deliberately provided "proof texts" from the Old Testament all throughout this epistle in order to "prove" his assertions or the doctrines which he was seeking to establish. Well, how then did the author "prove" that "the LORD shall judge HIS PEOPLE" or those who once belonged to Him? He "proved" the same by citing to his readers that which had previously been "written" in Deuteronomy 32:35-36 even as I cited the same above.
Anyhow, God's "vengeance" is not reserved solely for those who were always unbelievers, but also for those who once belonged to Him and then fell away. Scripture makes this plain throughout the Bible, yet multitudes reject it simply because they refuse to take God at His Own Word and interject their own biases into many passages of scripture such as the ones cited above.