I'll give my summary interpretation and then explain the reasoning behind it, so people don't need to read a long post just to get the interpretation:
The context of this passage is the Corinthians judging after the flesh, and the one example given is judging Paul after the flesh. They accused him of being powerful in letters but meek in person, thus a phony or hypocrite.
revenge all disobedience does not refer to punishing people, as the passage states "we do not war after the flesh".
Rather, it is the vain thoughts that exalt themselves above the glory of God that are being destroyed, not people. The people -- the church body -- are the ones being avenged, because they have been hurt by their vain thoughts. And really it is Christ in them that is being avenged, because Christ in them is the one blasphemed by judgments according to the flesh.
The vain imaginations are being destroyed by the obedience of the people, so that they are fully avenged when the obedience is complete.
This obedience will be caused by the Spirit moving on the hearts of the people, convicting them, in response to Paul's exhortations.
This is Paul's spiritual warfare: the preaching of the cross reveals the mystery that Christ is in them. Once the Corinithians understand that Christ is in the person they are judging, they realize that they are committing blasphemy, they repent, and they confess their own sins. This is how thoughts are taken captive, high places are pulled down, etc.
His weapons to accomplish this warfare are mighty and effectual even though the outward mechanism (preaching) is foolish and humble. Thus outwardly, he appears foolish and meek, but the warfare does not happen outwardly, but inwardly, where the Spirit moves in power.
"Having in a readiness" is most likely a Latinism -- "in promptu habere" which pictures a military display of readiness (to continue the military theme of the passage). Thus this passage is often translated as "standing ready to .." or "we stand ready" to help get this sense across of a military unit standing at attention, publicly displaying its state of readiness to some military leader that is reviewing it.
"we stand ready to revenge all disobedience, when your obedience is complete"
...should be explained as:
"We are a powerful army displaying our ability to avenge the vain imaginations of your hearts, by means of spiritual warfare, which will cause you to become fully obedient, confessing your sins, thus tearing down all your imaginations, and hence the body that was hurt will be avenged."
This passage is about judging wrongly, or judging after the flesh (according to what you can see with the eyes).
Remember that what matters is in the heart, because that's where Christ dwells. Every time you read Paul, keep in mind the mystery, the center of everything, what E.E. Cummings calls "the secret that nobody knows, the root of the root and the bud of the bud" -- that you carry another heart inside your heart, namely Christ in you, the hope of glory (Gal 1.26). There is no understanding Paul (or John's epistles) without first seeing the mystery.
Therefore because we are a temple, the only way to judge rightly is to see someone's heart - what is in the temple. Flesh cannot do that, it cannot comprehend Christ, it can only see flesh. So flesh always judges wrongly.
E.g. there are two people, one lusts after another man's wife in his heart, and the other commits adultery in the flesh. In God's eyes, they are both guilty of the same sin, since God sees both committing adultery in the Temple where He resides.
But in man's eyes, only one of these is judged. And it will be the one who committed adultery in the heart that will rise up to condemn the one who did it according to the flesh, because we only condemn in others what bothers our conscience in ourselves. That is, not seeing the heart, the carnal man projects his own heart onto others and then rises up to condemn.
Romans 2:1 (KJV 1900)
Therefore thou art inexcusable, O man, whosoever thou art that
judgest: for wherein thou judgest another, thou condemnest thyself;
for thou that judgest doest the same things.
Whereas the man whose conscience is pure does not condemn
To the pure all things are pure, but to those who are defiled and
unbelieving nothing is pure; but even their mind and conscience are
defiled. (Titus 1.15)
The Corinthians, which had no power to see into Paul's heart, nevertheless saw Paul's power in his letters, where he displayed his power, and they saw his meekness in the body. So they accused him of what they were guilty of -- only pretending to have power.
But when you judge outwardly, you are blaspheming God in the person you are judging. You are saying "You have problems in your temple". In that case, you better know what's happening in their temple, instead of judging by outward appearances.
Paul warns them
2 Corinthians 10:7 (KJV 1900)
7 Do ye look on things after the outward appearance? If any man trust
to himself that he is Christ’s, let him of himself think this again,
that, as he is Christ’s, even so are we Christ’s.
Thus these false judgements are an example of the imaginations that exalt themselves against the knowledge of God. They are what needs to be torn down. That is what is being talked about here.
Paul could have abused his power, and gone to the Corinthians, exposing them by revealing the secrets of their hearts - thus revealing their hypocrisy and that would tear everything down.
But Paul insists he uses his authority for edification, not destruction (2 Cor 10.8). So he wants them to tear down these imaginations by their obedience. That way they realize what they are doing and learn, as forced obedience is not obedience.
And the way to tear carnal judgement down is by confessing what troubles your conscience to the body, rather than projecting your guilt onto someone else and accusing them in the body. You lower yourself rather than exalt yourself.
But have renounced the hidden things of dishonesty, not walking in
craftiness, nor handling the word of God deceitfully; but by
manifestation of the truth commending ourselves to every man's
conscience in the sight of God. (2 Cor 4.2)
When you openly confess what your conscience troubles you about, then you will not judge another wrongly. Then you will, like Paul, resolve never to judge after the flesh:
Wherefore henceforth know we no man after the flesh: yea, though we
have known Christ after the flesh, yet now henceforth know we him no
more. 2 Cor 5.16
Thus Paul will exhort them to confess. He will teach them the mystery and he will preach the cross. This is how he wages spiritual warfare. And through their obedience, the imaginations will be torn down, and those who have been blasphemed - namely the church body - will be avenged.