Did James and Jude Write under the Oversight of some Apostles?
What Criteria Did the Early Church Use to Identify Writings of the NT?
The question raises a few questions of its own, for instance, what criteria are being used (or are thought to have been used) by the early church to recognize and identify (not codify, confirm, or assent) the Holy Spirit-inspired writings of the NT? Was there a checklist-like system where if someone besides an Apostle wrote, it was immediately rejected as being uninspired/not from the Lord? That might make some of our understanding of the Book of Hebrews very much "up in the air". It becomes a bit of a cyclical argument "Hebrews must have been written by an Apostle because only writings of the Apostle's were recognized as inspired". So the Apostolic origin theory is a bit heavy to lean on.
We also know there must be some more criteria to recognize the scriptures as inspired beyond merely written compositions from Apostolic writers. Paul cites a few additional writings (particularly to the Corinthian church) that show not everything penned by the Apostles was inspired as some of those correspondence letters have not been preserved by God or recognized by the church as inspired.
So, we come to Jude and James and we go....did they HAVE to have been under Apostolic oversight in the personal sense (as an editor, amanuensis, or final publisher)? My answer, and I believe I stand on firm historical and systematic ground to say "No".
In another similar vein, would we say that Jude and James came under Apostolic oversight in the general sense of the content of their teaching/message/written letters comported with (was in copacetic relationship to) the Apostle's teaching? YES. Nothing in the scriptures is contradictory in such a way as to erode, take away from, or diminish the truth, or authority of God.
Holy Spirit Document
In this second sense, we can avoid any pitfalls of merely human ideas of Apostolic succession or divine speaking in every utterance or writing from the Apostles (which we humans are often prone to do in the building up and making of spiritual heroes into idols) and avoid the error that it is only through a mere rational checklist of present philosophical grids that the NT was recognized and composed. It is a Holy Spirit document, living and active, and not subject in any way to the measurement or standards of any time or culture to demonstrate its authentic authority and purpose.
Many people are genuinely uncomfortable with Holy Spirit language, but that is what Peter, Hebrews, Paul, and Jude affirm. The distinctive messages that come from God are by the Spirit through the prophets in the previous days, and through the Son by the Spirit in these last days.
Personal Relations
Lastly, we need to be careful to make exceptions stating that it is James or Jude's personal relations or close shared life experience with Jesus that acted as authenticators of their writing. Would this then have given rise and allowed for others to claim to have been witnesses or near Christ and speak in a manner contrary to the Apostolic teaching and witness? It certainly did in history! That's one of the reasons for so many of the false gospels and early gnostic writers to deploy pseudepigrapha (false authorship claims) status as authors.
Neither James nor Jude make special claims regarding their message as a result of their earthly affiliation or closeness to Jesus. There was not going to be a religious familial hegemony attempted based on the physical/earthly lineage of those related to Christ. The writers of James and Jude both write in such a way as to *almost* distance themselves from personal close earthly ties to Christ to dissuade any from following them rather than following our Lord. This has been confirmed throughout particularly modern critical history as plenty of various alternative authorship theories have been given to these books. That is NOT to say that Jude or James were written by anyone else besides...Jude and James. That is to say, the manner in which they wrote allowed for their own authority and relationship to be undercut, and even if it was, the message and truth of God's Word given to them to relay through their written communication still shines through to those whom God gives eyes to see and ears to hear by His Holy Spirit.
Article 5 of the Belgic Confession of Faith
“We receive all these books and these only as holy and canonical, for the regulating, founding, and establishing of our faith. And we believe without a doubt all things contained in them—not so much because the church receives and approves them as such but above all because the Holy Spirit testifies in our hearts that they are from God, and also because they prove themselves to be from God. For even the blind themselves are able to see that the things predicted in them do happen.”
I hope some of these thoughts are helpful to you in your study. As with all things canonical, Michael Kruger is an excellent resource for further study/discussion.