I'm not sure. From TOS onward, we no longer see any indication that the Vulcans, Tellarites, and Andorians had their own fleets, at least non-Earth style vessels. We know that the Vulcans had transport craft, but it generally doesn't seem to make much sense that each of these worlds would continue to build their own ships after the founding of the Federation. It makes more sense to have one singular Starfleet charged with exploration and defense of the various member worlds/systems.
I don't see how that makes sense, because the Federation is just that, a confederation of sovereign nations cooperating for mutual benefit and defense. It's not the United States writ large. It's more like the European Union (though not exactly, just relatively). If they still consider themselves sovereign nations, it stands to reason that they'd want to be responsible for their own defense and transportation capabilities as well as cooperating in mutual defense.
And we do have evidence of one all-Vulcan starship, the Intrepid
. Fans have assumed for decades that it was one of the Constitution
-class fleet, and there's behind-the-scenes material reinforcing that assumption, but the bottom line is, we never actually saw the ship. It could've been a Vulcan design. Also, there's the Vulcan-made shuttle and warp sled in TMP, although the shuttle portion conforms to familiar Starfleet designs.
I think there was some indication of Vulcan having its own capital ships in "Unification" or "Gambit" or some such episode. Once at a Trek museum exhibit, I saw a Rick Sternbach design drawing for a TNG-era Vulcan starship, something that had familiar Starfleet technology elements but a more Vulcan design overall. It was designed for one of those episodes, I think, but ultimately not shown.