Is it possible to believe the Bible is inerrant if you don't have faith?
If by faith you mean "believing" and "embracing" the Bible then, no, it is not possible. The Devil knows that the Bible is inerrant, but it doesn't do him any good. To believe that it is inerrant and to embrace it as such is only possible with faith.
But faith is not a blind leap. Faith doesn't come to Bible and say, "Well I don't know what you say, Bible, but I'm going to believe you." That is not an honor to the Bible or to God.
Faith begins to read the Bible, and the eyes of the heart are opened to see God and to see Christ portrayed so that the self-authenticating power of the word draws you to close with it and causes you to believe that Paul is inspired and that Jesus is who he says he is.
Then you are left with all of these other pages in the Bible that you haven't worked with yet; and as you start reading through them you begin to bump into issues (or someone points them out to you). What faith does in those times is hold fast to what it has seen in the word, and then it attempts, without throwing away the whole Bible, to embrace the whole Bible with understanding.
So faith does have to suspend condemnation when it bumps into a problem that it cannot immediately solve.
This happens all the time in our devotional readings (if we're finite people and haven't studied the whole Bible fully and completely). You're reading along and some character in the Bible says something that makes you ask, "How does that fit with what I saw over here?"
If you have the time you can go to work on it by getting out commentaries, etc. My experience is that if I have the time I can generally come to some resolution or some plausible conviction about the passage.
But most people don't have the time or the wherewithal for a more in depth study. At that point, faith in what they have seen in Jesus and in the word will enable them to hold fast and say, "I may not have the solution here, but I believe there is a solution, and someday I'll see it."
So yes, it takes some measure of faith to believe and embrace the inerrancy of Scripture. But it's not blind. In fact, it's rooted most profoundly in seeing, according to 2 Corinthians 4:4-6 especially:
In their case the god of this world has blinded the minds of the unbelievers, to keep them from seeing the light of the gospel of the glory of Christ, who is the image of God. 5 For what we proclaim is not ourselves, but Jesus Christ as Lord, with ourselves as your servants for Jesus’ sake. 6 For God, who said, “Let light shine out of darkness,” has shone in our hearts to give the light of the knowledge of the glory of God in the face of Jesus Christ.