In the episode "wej Duj", Season 2 Episode 9 of the Star Trek: Lower Decks series that takes place in the late 24th century and which is considered canon (*), we see the Vulcan Cruiser Sh'val, which is non-Starfleet and is crewed entirely by Vulcans.
At the end of the episode, the young Vulcan who is the focus of the episode (T’Lyn) gets a transfer to Starfleet as apparently she was too rebellious for the Vulcan fleet.
Hence, at least one of the species, namely the Vulcans, still have their own fleet independently from Starfleet.
(*) The show's canonicity was confirmed by its creator Mike McMahan in a recent interview:
So the trick with ‘Lower Decks’ is that our characters are, our stories and our characters are definitely in canon.
As this info has not been contradicted by CBS thus far, it's safe to assume it still stands.