What is the Old Testament Biblical basis to the idea that only God can forgive sins?

Under the Jewish theocracy, wilful disregard of the positive, or wilful infraction of the negative, commands of God as proclaimed by Moses and interpreted by the Rabbis; it thus includes crimes against God and crimes against society or an individual member thereof.
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The earliest Biblical conception of what constituted sin is illustrated by the story of Adam's punishment, which was due to his failure to obey the divine will and his revolt against the divine government. The catastrophe of the Flood was a punishment for man's demoralization and corruption, his violence and immorality (see Gen. vi. 11, 12). The builders of the Tower of Babel revolted against divine government, and were dispersed (see Gen. xi. 1-9). Sodom and Gomorrah were destroyed for their heinous crimes: "The men of Sodom were wicked and sinners before the Lord exceedingly" (Gen. xiii. 13): they were "wicked" in civil matters, "sinners" in blasphemy "exceedingly," with full appreciation of the enormity of their sins (Sanh. 109a). The Egyptians were punished for the sin of enslaving the Israelites, and for not heeding the command of God to release them. The most serious sin of the Israelites was the worship of the golden calf, contrary to God's commandments delivered from Sinai. Korah rebelled against the authority of Moses, and of the Levites, priests by the choice of God. The Canaanites practised incest and immorality: "For they committed all these things, and therefore I abhorred them" (Lev. xx. 23); "But for the wickedness of these nations the Lord thy God doth drive them out from before thee" (Dout. ix. 5).
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Jewish theology does not admit that there is an unpardonable sin. The Mishnah says that sins are expiated (1) by sacrifice, (2) by repentance at death or on Yom Kippur, (3) in the case of the lighter transgressions of the positive or negative precepts, by repentance at any time.
— SIN - JewishEncyclopedia.com

The dominant theme here is that sin is willful disobedience of God's commandments.

As 1 John 3 explains it in the New Testament:

Whosoever committeth sin transgresseth also the law: for sin is the transgression of the law.

Sin is disrespecting God. The idea that anyone other than God himself could forgive direct attacks against him just doesn't make sense.

Suppose someone did something awful against you. Would it make sense for me to forgive them?

Would you ever consider delegating someone else to offer forgiveness every time someone hurts you?

Or consider the reverse analogy. Would it ever make sense to delegate your ability to love someone to a third person?

Specifically one of the best Old Testament verses per your request is Jeremiah 31:34. "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."

When do you think this "prophecy/covenant" took place? John 1:29, "The next day he/John the Baptist saw Jesus coming to him, and said, "Behold, the Lamb of God who takes away the sin of the world."

Notice "the sin" (ten hamartian is interesting because of the definite article and the singular number. This speaks of the root source and nature of sin, not merely of the symptoms called sins (plural). And yes, "the sin" may also refer to the totality of the sins of the world in a collective sense.

Now, you added the following "caveat," "and not one who is delegated with authority by God?" If your know or if your already convinced that only God can forgive sins why bother to say, "don't bother with some person delegated with authority by God?" What purpose does it serve?

In fact, if you think about it every single thing Jesus did was with the approval of His Father, and that includes not only dying for the sins of the world but forgiving sins as well. What does Luke say at Luke 9:35? "And a voice came out of the cloud saying, "This is My Son, My Chosen One,(or if you will My delegate) LISTEN TO HIM."

Finally, 2 Corinthians 6:15-16 the Apostle Paul refers to Christ at vs15 and vs16 says, "I will dwell in them and walk among them; And I will be their God, and they shall be My people." Who was the person that literally walked and lived among the people? Who made their abode or lived IN people? John 14:23 as well as John 14:16.

What is the Old Testament Biblical basis to the idea that only God can forgive sins?

God’s forgiveness is redemptive in character.

Israel, put your hope in the Lord, for with the Lord is unfailing love and with him is full redemption. He himself will redeem Israel from all their sins. - Psalm 130:7-8

God’s forgiveness is transformational.

Come now, let us reason together," says the LORD. "Though your sins are like scarlet, they shall be as white as snow; though they are red as crimson, they shall be like wool. - Isaiah 1:18

God moves our sins out of His sight.

Surely it was for my benefit that I suffered such anguish. In your love you kept me from the pit of destruction; you have put all my sins behind your back. - Isaiah 38:17

God chooses not to remember our sins anymore.

I, even I, am he who blots out your transgressions, for my own sake, and remembers your sins no more. - Isaiah 43:25

God dissipates our sins forever.

I have swept away your offenses like a cloud, your sins like the morning mist. Return to me, for I have redeemed you. - Isaiah 44:22

Yet, in a way, I can understand why the Scribes and Pharisees thought that Jesus was speaking blasphemy. For them Jesus is not the Messiah or the Christ, the Anointed One of God. They are unbelievers.

But the Scribes and Pharisees began thinking to themselves, “Who is this man who speaks blasphemy? Who can forgive sins but God alone?” - Luke 5:21

God’s forgiveness does not dredge up our sins anymore.

For I will forgive their wickedness and will remember their sins no more. - Jeremiah 31:34

God’s forgiveness pardons sinners.

In accordance with your great love, forgive the sin of these people, just as you have pardoned them from the time they left Egypt until now." The LORD replied, "I have forgiven them, as you asked. Nevertheless, as surely as I live and as surely as the glory of the LORD fills the whole earth. - Numbers 14:19-21

God forgiveness is purifying at its core.

Wash away all my iniquity and cleanse me from my sin… Cleanse me with hyssop, and I will be clean; wash me, and I will be whiter than snow. - Psalm 51: 2; 7

God’s forgiveness is cleansing in nature.

…on this day atonement will be made for you, to cleanse you. Then, before the Lord, you will be clean from all your sins. - Leviticus 16:30

Ultimately it is only God who can forgive sins. But in some mysterious way He is able to do so through his High Priests. Only God can forgive sins. But at the same time, it makes sense that God delegated that authority to His Old Testament priests. Mankind, in this instance, the Ancient Jewish Nation needs some reassurance from the Almighty that their sins can be forgiven.

The closest Old Testament biblical support for this would be from Numbers 15:25:

And the priest shall make an atonement for all the congregation of the children of Israel, and it shall be forgiven them; for it is ignorance: and they shall bring their offering, a sacrifice made by fire unto the LORD, and their sin offering before the LORD, for their ignorance (KJV).

As Christ made atonement for our sins on the Cross, in the New Covenant, so too in a prefigured manner, the High Priests of the Ancient Covenant made atonement for the sins of the people of the Israel nation, God’s Chosen People once a year.

St. Paul in Hebrews 9:7 speaks as thus:

But only the high priest entered the second room, and then only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance.

I am going to expand on the excellent points made in Ken Graham & Ray Butterworth's answer into a unified understanding.

In your comment to Ken Graham's answer you asked:

Are you saying there is no OT basis to the idea that only God can forgive sins, and that the scribes and Pharisees were simply incorrect?

The short answer is that the Pharisees of course recognized the concept of delegation through God-provisioned priesthood operating with the ritual sacrifices in the single authorized place: the temple in Jerusalem. When they say "Who can forgive sins but God alone?" they implicitly mean "Who can forgive sins but God (and his authorized priests) alone?" thus rejecting Christians's later claim that Jesus is our eternal High Priest, as well as previously rejecting John the Baptist's claim to be acting as God's agent to forgive sin (see Mark 1:4).

Let's take it step by step:

1. Only God can forgive sin since ultimately all infractions are sin against Him. Ray Butterworth's answer provides good OT context for this based on modern Judaism's own understanding.

2. But in the OT God provided authorized representatives (the priesthood) who can also forgive sins through sacrifices. Ken Graham's answer provides good OT context for this "channel" of forgiveness.

3. In the NT God provides Jesus as the eternal High Priest but the Pharisees didn't recognize Jesus for this role. Thus, Mark 2:7 and Luke 5:21 where they say "Who can forgive sins but God alone". But they didn't reject the concept of delegation (see my short answer above). They just didn't recognize Jesus as God's authorized representative.

4. In Catholicism the Catholic priests are delegated authority by the High Priest Jesus for the same function. Thus we have Catholic priests who can legitimately say "I absolve you from your sins" in the Sacrament of Penance just as Jesus also said "Son, your sins are forgiven.". Full text from the sacrament:

   God, the Father of mercies, through the death and resurrection of his Son has reconciled the world to himself and sent the Holy Spirit among us for the forgiveness of sins. Through the ministry of the Church may God give you pardon and peace. And I absolve you from your sins in the name of the Father, and of the Son, and of the Holy Spirit.

Conclusion

Yes, only God can forgive sins. But at the same time, it makes sense that God delegated that authority to the OT priests, to Jesus, and to His apostles and successors. I realize that not all Christians agree with this interpretation; see here how the same verses used by Catholics for this delegation are interpreted differently by Protestants.

In setting out to demonstrate that only God can forgive sin, the first premise to establish is that all sin is primarily or ultimately against God. The premise in question (the Old Testament Biblical basis to the claim that God alone can forgive sins, and not one who is delegated with authority by God) follows naturally after that.

When David committed adultery with Bathsheba and murdered her husband Uriah (2 Samuel ch. 11-12) , and after David is confronted by Nathan the prophet, David says: "I have sinned against the Lord".

Psalm 51 states that it is written by David and has the Bathsheba incident and God's sending of Nathan the prophet as it's subject matter:

To the choirmaster. A Psalm of David, when Nathan the prophet went to him, after he had gone in to Bathsheba. - Psalm 51:1

What David declares in this Psalm is:

Against you, you only, have I sinned and done what is evil in your sight, so that you may be justified in your words and blameless in your judgment. - Psalm 51:4

This is not to ignore the sins against Bathsheba and Uriah but to indicate that the primary victim of all sin is God.

In the Levitical law a wrong done to another was required to be recompensed to that other, for instance:

If anyone sins and commits a breach of faith against the LORD by deceiving his neighbor in a matter of deposit or security, or through robbery, or if he has oppressed his neighbor or has found something lost and lied about it, swearing falsely—in any of all the things that people do and sin thereby— if he has sinned and has realized his guilt and will restore what he took by robbery or what he got by oppression or the deposit that was committed to him or the lost thing that he found  or anything about which he has sworn falsely, he shall restore it in full and shall add a fifth to it, and give it to him to whom it belongs on the day he realizes his guilt. - Leviticus 6:2-5

But notice that this deception against the neighbor is classified primarily as a breach of faith against the Lord. He must make reparation (with interest) to his neighbor and then he must bring the appropriate offering to the temple to apply for forgiveness from the Lord through the sacrificial system that He had instituted for that purpose (Matthew 5:23-24).

And he shall bring his trespass offering unto the LORD, a ram without blemish out of the flock, with thy estimation, for a trespass offering, unto the priest: And the priest shall make an atonement for him before the LORD: and it shall be forgiven him for any thing of all that he hath done in trespassing therein. - Leviticus 6:6-7

In this way, restitution or reparation to a neighbor for wrong done is a requirement for the appeal to God for forgiveness. It is a tangible demonstration of repentance and as much a part of an appeal to God as the trespass offering brought to the priest in the temple. It is in this spirit that John the Baptist chastised the Pharisees and Sadducees to "bring forth fruit in keeping with repentance" (Matthew 3:7-9).

God had established the temple and the priesthood as the means by which forgiveness from God could be obtained. And that is the salient point. Forgiveness for sin is sought from God by the means that God has established because all sin is against God. One may apply to God for forgiveness by the means dictated but, it is God (and not the means themselves (Hebrews 10:4)) who forgives. God may deliver forgiveness apart from application through this process (but not absent repentance) but that is his prerogative and not to be demanded or expected by us.

And David said unto Nathan, I have sinned against the LORD. And Nathan said unto David, The LORD also hath put away thy sin; thou shalt not die. - 2 Samuel 12:13

And so Nathan has declared that David is forgiven but the declaration is that the Lord has forgiven. Nathan has not forgiven; he has pronounced the Lord's forgiveness. Likewise the priests in the temple receive application to God by means of sacrifice and confer upon the penitent the Lord's forgiveness. The priest does not forgive; he mediates/procures the Lord's forgiveness. The priest and the prophet are part of the means.

There was a levitical process for a leper to avail himself for healing/forgiveness. One leper bypassed the temple, the priests, and the offerings and went straight to Jesus. When Jesus healed the leper he commanded the man to go to the priests and offer the appropriate sacrifice:

And it came to pass, when he was in a certain city, behold a man full of leprosy: who seeing Jesus fell on his face, and besought him, saying, Lord, if thou wilt, thou canst make me clean. And he put forth his hand, and touched him, saying, I will: be thou clean. And immediately the leprosy departed from him. And he charged him to tell no man: but go, and shew thyself to the priest, and offer for thy cleansing, according as Moses commanded, for a testimony unto them. - Luke 5:12-14

He commands him thus not as an appeal to God for healing but as a testimony to the priests that God's healing took place by the hand of Jesus.

Later, when the paralytic was lowered down to Jesus by his friends this was also done in an environment wherein the temple system was still intact. If healing was sought there was a levitical process in place and specific sacrifice and ritual already in place. When Jesus declared to him "Man, your sins are forgiven you" he was taking the place of the entire Levitical system.

The Pharisees and the teachers of the Law were offended because Jesus pronounced forgiveness himself. He did not proclaim, as the prophets or priests would do, that the Lord had forgiven the man. He pronounced the man "as forgiven" quite apart from the process to apply for forgiveness that God had set in place. He did that which it was only God's prerogative to do and the miracle of healing was confirmation of the same:

And the scribes and the Pharisees began to question, saying, “Who is this who speaks blasphemies? Who can forgive sins but God alone?” When Jesus perceived their thoughts, he answered them, “Why do you question in your hearts? Which is easier, to say, ‘Your sins are forgiven you,’ or to say, ‘Rise and walk’? But that you may know that the Son of Man has authority on earth to forgive sins”—he said to the man who was paralyzed—“I say to you, rise, pick up your bed and go home.” - Luke 5:21-24

This authority is rightly "power" or "ability" and it is not authority to declare God's forgiveness of sin (as prophet and priest might do) but it is authority to forgive sin (as only God can do). This is exactly what the Pharisees recognized and reacted against, “Who is this who speaks blasphemies? Who can forgive sins but God alone?”

Has is the verb εχω (echo) and means to have or hold. Its usage slightly transcends that of the English equivalent. Where our English verb "to have" mostly emphasizes possession and control of external things, our Greek verb mostly describes the set of features that collectively define identity. When someone "has" something (say: an object or property, a quality or condition, a word to say, a feeling, a skill, an obligation or conviction), that something helps to determine who that person is. The Greek verb εχω (echo) sums up the substance of one's reality, it sums up the internal and external qualities of someone or something rather than merely the external things that come in addition to someone or something.

When Jesus declared that the Son of Man "has authority on earth to forgive sins" he is not saying that he wields a power or holds an office that was conferred upon him. He is saying that he, himself, is forgiveness and, since only God can forgive sins, He is God's forgiveness.