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Not every book of the New Testament was written by one of the apostles. Scripture itself shows that God also used close companions and coworkers of the apostles to record inspired teaching. These writers were trusted because they faithfully preserved apostolic doctrine.

Below are the New Testament books traditionally understood to be written by non-apostles, along with the biblical reasons.

Books Written by Non-Apostles
Mark — Gospel of Mark
Mark was not one of the Twelve but was closely connected to Peter.
1 Peter 5:13 “Marcus my son”
Early testimony says Mark recorded Peter’s preaching

Luke - Gospel of Luke & Acts
Luke states plainly that he was not an eyewitness, but carefully investigated the accounts.
Luke 1:1–4
Colossians 4:14
2 Timothy 4:11
Luke was a close companion of Paul.

James - Epistle of James
James identifies himself as a servant of God and of the Lord Jesus Christ.
James 1:1
Galatians 1:19 — “James, the Lord’s brother”
He was not one of the Twelve, yet a recognized leader in Jerusalem.

Jude - Epistle of Jude
Jude identifies himself as the brother of James.
Jude 1:1
This places him among Jesus’ family (Matthew 13:55), not among the Twelve.

Hebrews - Author Not Named
Hebrews does not name its author and does not claim direct apostolic authority.
Hebrews 2:3 — “confirmed to us by those who heard Him”
Hebrews 13:22–23 — sounds like a spoken exhortation and mentions Timothy.
Many have debated its authorship.
Personally, I believe Paul wrote Hebrews, or that it originated from his teaching (possibly as a sermon later written down), because no one else in the New Testament shows the same depth of understanding of the Law, priesthood, and Christ’s fulfillment of them. Still, Scripture itself does not name the author.

Summary
Books of the New Testament not written by one of the Twelve apostles:
Mark
Luke
James
Jude
Hebrews (author unnamed) However, I would say Apostle Paul

They were accepted because they faithfully preserved apostolic doctrine, not because every author held the title “apostle.”