I don't know what you might be referring to that leaves some ambiguity on this point.
JTB pointed out the USS thing, so I won't argue for non-Starfleet any more. But if I did, here's the lowdown:
The star base Enterprise was heading towards, orders them to look into the ship's disappearance, along with the system it was investigating. I don't think such a description would be used if the Intrepid was not a Starfleet vessel.
Why not? Kirk was often ordered to look after non-Starfleet vessels (Mudd's, Sevrin's, all sorts of unidentified vessels), and systems that were not military bases. SB6 was in "contact" with the
Intrepid and with the Gamma VII system, and one of those was decidedly not a Starfleet asset!
Spock refers to the death of 400 crew members, a deliberate nod, I think to another Constitution-class ship.
Oh, agreed very much. But that does nothing to remove ambiguity on screen. Indeed, we see that Spock is the person who knows the
Intrepid and her 400 Vulcans, with none of the other heroes indicating familiarity until Spock is done with the exposition. She might well be a rather random Vulcan vessel, then. Even if famous in the Vulcan circles.
If it wasn't Starfleet, under whose aegis would it have been operating?
Why should that matter? Vulcan Science Academy, perhaps. Or American Continent Institute, for all we care. Trek is rife with possibilities and precedent.
If it was a Vulcan ship , why on earth would it be called the Intrepid?
Why not? Vulcans aren't prone to trepidation, or at least won't admit to it.
For what it's worth, MA states unequivocally that the ship, was unambiguously of Starfleet issue.
Then it's time to edit Memory "Often Wrong" Alpha.

Really, we're the masters of that resource, not vice versa.
Doesn't mean we should declare a matter settled if it is not. Fans creating
Intrepid insignia and kit decals won't alter what MA should say, any more than fans "generally agreeing" that Spock was the first Vulcan in Starfleet would make that claim worthy of repeating in MA.
Timo Saloniemi