Let's consider a comparison from the entire scope of the Old Testament.
The OT itself is a story of God's progressively revealing what He wanted to His people when He wanted. When a prophet speaks, you are obligated to listen, simple as that. Generations of believing Hebrews lived and died with only Genesis in oral tradition, then later generations with only Genesis in writing, then more generations with only Genesis in writing and some of Exodus, Leviticus, and Numbers orally passed down. More and more progressively committed to writing, and yet other oral traditions - not inspired - being passed down as well, like 2 Kings 1:18 - "are they not written in the book of the annals of the kings of Israel?", an inoffensive one like Hanukkah, or an anti-biblical one like that condemned by Jesus in Mark 7:1-13. Two categories - Scripture and not-Scripture.
God sets the time for speaking, God provides for His people before, during, and after, in His way. Who is the man who will put God in the dock and say "Why have You done this?"
A specific example - King Josiah 2 Kings 22.
The point is simple - when God decides to send His Word to His people, we trust Him to make it known, and to provide for its communication to the people to whom He wanted to communicate it, all in His timing. Just because it's messier than we might like it doesn't mean we are justified in ignoring it.
The challenge for the questioner, then, is to demonstrate that there is indeed another source of divine revelation besides Scripture.
V 3 - You want me to act? I will, but in my own good time.
Cf 2 Peter 3:8-9, Isaiah 46:9-11
Is God's action any less certain just because it takes longer than we think it will or should? How pathetic does that make me because of my lack of trust in Him, that I think of things on the basis of whether I see an answer in an hour or day?
v 4 - The man who trusts God will live and be righteous. He who would call God on the carpet for not acting fast enough is proud, proud enough to correct Almighty God! And his soul is obviously not properly oriented toward God. Or do you think you know how to run the world better than He does?
So, speaking of which, how were people in the OT saved?
Notice that God is telling Habakkuk to trust Him that the Babylonians will be dealt with. But whom was He telling Hab just last answer that He'd be dealing with? Judah! It's not one or the other, as we often like to try and set up as a dilemma. "God, don't deal with ME! I'm not as bad as THAT GUY!"
No, God deals with ALL sin, ALL lawbreaking, when He has decided to. Not some of it, not only the really really bad stuff. It's a both-and.
And God's answer to Habakkuk is to take a wider perspective: don't try to tell Me how to run the world and enact justice - you should be more interested in living, and living is by faith. "What good will it be for a man if he gains the whole world, yet forfeits his soul? Or what can a man give in exchange for his soul?" - Matt 16:26
v 5 - Like death itself, which doesn't stop until it takes 100% of life, even good men, little children, pastors, etc, these Babylonians are not satisfied with "home", ie, what God has given to them as their portion?
Nebuchadnezzar of Daniel 4:28-30 - why does this king, this mortal man, think it is permissible for him to assemble a violent army to kill a bunch of people and take their stuff? What does Neb say as to the reason?
And how is this any different from the modern idolater? Tycoon banker? Drunkard? Senator? Drunkard Senator? Gambling addict whose family loses out to the addiction?
v 6 - the lesson is that your deeds will find you out, will bite back. God will arrange punishments and sufferings for the Babylonians to endure as well. Further, He'll arrange for them to be mocked and scorned. Think about it - isn't the end of King Belteshazzar of Daniel 5 pretty pathetic and worthy of all the contempt and scorn heaped upon it for centuries? Especially the way he blows off the obvious miraculous sign, failure of his normal mystical aids, and accompanying prophecy? Only to be destroyed that very night?
BONUS if we have time: Exegesis of Acts 13:13-41, in which Paul quotes Habakkuk 1:5.