Evangelical view of the inerrancy of Bible does not necessarily imply strict apostolic authorship, but rather the certainty that:
Being wholly and verbally God-given, Scripture is without error or fault in all its teaching, no less in what it states about God's acts in creation, about the events of world history, and about its own literary origins under God, than in its witness to God's saving grace in individual lives.
(Short Statement #4 in the 1978 Chicago Statement on Biblical Inerrancy signed by more than 200 evangelical leaders (see wikipedia).
Even B.B. Warfield, a key figure in the evangelical doctrine of inerrancy, wrote in his 1892 article The Formation of the Canon of The New Testament (emphasis mine):
Let it, however, be clearly understood that it was not exactly apostolic authorship which in the estimation of the earliest churches, constituted a book a portion of the “canon.” Apostolic authorship was, indeed, early confounded with canonicity. It was doubt as to the apostolic authorship of Hebrews, in the West, and of James and Jude, apparently, which underlay the slowness of the inclusion of these books in the “canon” of certain churches. But from the beginning it was not so. The principle of canonicity was not apostolic authorship, but imposition by the apostles as “law.” Hence Tertullian’s name for the “canon” is “instrumentum”; and he speaks of the Old and New Instrument as we would of the Old and New Testament. That the apostles so imposed the Old Testament on the churches which they founded — as their “Instrument,” or “Law,” or “Canon” — can be denied by none. And in imposing new books on the same churches, by the same apostolical authority, they did not confine themselves to books of their own composition. It is the Gospel according to Luke, a man who was not an apostle, which Paul parallels in I Tim. v. 18 with Deuteronomy as equally “Scripture” with it in the first extant quotation of a New Testament book of as Scripture. The Gospels which constituted the first division of the New Books, — of “The Gospel and the Apostles,” — Justin tells us, were “written by the apostles and their companions.” The authority of the apostles, as by divine appointment founders of the church, was embodied in whatever books they imposed on the church as law, not merely in those they themselves had written.
For more details on how Evangelicals grappled with inerrancy of the Bible and the rather fluid canon formation in the early church, see a good introductory article Evangelicals and the Canon of the New Testament. This talks about how Warfield's distinction specifically relates to Hebrews:
The practical effect of this subtle distinction is to allow for the inclusion of books such as Mark, Luke, James, Jude and Hebrews which were not actually penned by the apostles, but were, according to tradition, written under apostolic sanction.
For a 2010 conservative evangelical attempt to establish the authorship of the Book of Hebrews, see Brian Wagner's paper Another Look at the Authorship of Hebrews from an Evangelical Perspective of Church History published in the Journal of Dispensational Theology, where he "decisively stated" that Barnabas was the author. This should make the Book of Hebrews very close to Pauline authority, which in my opinion should make conservative evangelicals believing in inerrancy to be much more comfortable.