The releasing of the azazel into the wilderness (Leviticus 20:21-22) is commonly—but incongruously—linked with the binding of Satan (Revelation 20:1-3). This linkage is problematic because the account in Revelation makes no mention of sins—whether mankind's or Satan's—being placed on Satan's head. Instead, the express purpose for his binding is “so that he should deceive the nations no more till the thousand years were finished” (verse 3). He is bound to cut off his influence, not to provide any sort of expiation.
In addition, since the book of Revelation is generally dated around AD 90-100, John's vision did not occur until the very end of the apostolic generation. Earlier, Jude had written to exhort the beleaguered first-century church “to contend earnestly for the faith which was once for all delivered to the saints” (Jude 3; emphasis ours throughout). “The faith” is a specific faith, indicating a well-defined body of beliefs. However, when Jude wrote his epistle, Satan's binding had not yet been foretold. It was not revealed to John until after “the faith” had already been delivered to the saints.
Similarly, Paul writes to Timothy that the “Holy Scriptures”—which, at that point, would have consisted of what is generally called the Old Testament—“are able to make you wise for salvation through faith which is in Christ Jesus” (II Timothy 3:15). Even without the book of Revelation, the first generation of Christians had all they needed to understand God's plan.
The matter of how sin is atoned for is at the very core of being “wise for salvation,” and Paul's words indicate that “wisdom” was already available from 1) what had been written before, and 2) God-given faith in Jesus Christ. In other words, the apostolic generation understood salvation—including the Day of Atonement and how sins are expiated—without the revelation that Satan will be bound. The former is not dependent on the latter.