As far as onscreen material is concerned, Starfleet does seem to be the Federation's only armed uniform service. Yes, I'm tapdancing around the M word since I don't want this to turn into one of Those Threads. Whatever else we think Starfleet may or may not be, we can all agree it is a uniform service and that it is armed.
Anyway, on topic, we see constantly, Starfleet is the only Federation force to respond to threats, be that the Borg or the Dominion or Klingons or whoever we've seen the Federation at war with. Betazed fell because of Starfleet's inaction, when Bajor was going to join the Federation they were even going to "absorb" the Bajoran Militia into Starfleet. And these days we know the MACOs were disbanded when the Federation was founded.
Though it is interesting to note, our view on this is based mostly on what is seen in the Berman era, which arguably comprises the majority of on-screen material. TOS certainly implies Federation member worlds had their own forces. After all, in Journey to Babel the delegates were all from Federation worlds which were ready to go to war with each other. War would be difficult without your own M word, especially if everyone is relying on the same exploration service which is allegedly not an M word for defense. And in Disco we have the aforementioned Vulcan Expeditionary Group. One interesting thing from Enterprise, while looking through Mr. Daniels's database, T'Pol points to a Vulcan cruiser she doesn't recognize and Archer says it hasn't been built yet, pointing to a date. Unless that cruiser is built within the next decade, it would almost certainly be a post-Federation founding Vulcan ship.
Then, there's DS9 which intentionally avoided mentioning anyone other than Starfleet or the Federation government in the Homefront/Paradise Lost story fearing it would confuse the audience. Then they took it too far in the season 7 premiere and actually had Starfleet Security and Medical handle the matter when Sisko is attacked by the Pah-Wraith cultist in New Orleans. Civilian law enforcement on Earth isn't even seen in the franchise at all until Trek XI, and there were those who actually cried at that being a canon violation. Even during Enterprise, the only prison mentioned on Earth is a Starfleet-run one. Though some sources do say the agents who arrest and detain McCoy in Trek III were civilian law enforcement, it should be noted the jail guards do have Starfleet arrowheads on their uniforms.
The novels do posit that Federation worlds do maintain their own defense forces to deal with local affairs, the Bajoran Militia continues to exist after Bajor joins the Federation, and the Andorian Imperial Guard does still exist in the 24th century. Also that there is civilian law enforcement on all Federation worlds, even Earth, as well as a federal agency that has jurisdiction across the entire Federation.