Watching ST:VOY S05E15, Dark Frontiers (pt 1), we get to see flashbacks of Seven of Nine's parents while they were aboard the Raven, which appears to be a very Federation design (nacelle placement notwithstanding). How did they get their ship? How do any civilians in the Federation (ie Cassidy from DS9) get their ships? Can I just get on a waiting list?
Since this answer has gotten more notice. I invite you to read Rick Webb's "The Economics of Star Trek".
If you went to a replicator, or a dealer, or the Utopia Planatia Fleet Yards and asked for 10 million star ships, the answer would be no. More concretely, when the Borg attacked, and during the Dominion War, the Federation suffered from a serious starship shortage.
There is some level of scarcity — the Federation cannot manufacture a million starships, for example.
As mentioned in the comments. One way is to petition the Federation for a ship for a certain assignment, such as USS Raven for exploration of reports of the Borg species. Presumably had they survived and completed their mission, they would have turned that ship back over to the Federation when they returned.
There are a number of transport ships (several like the one Cassidy has) that are operated by civilians. These could be the equivalent of Merchant Marines, where the Federation is supplying transport and cargo ships to civilians who want to operate them.
We also know that eventually the Federation will decommission ships and place them into a depot. From there they salvage any useful (i.e. modern?) tech and then the ship goes into storage. You see a similar place encountered by Voyager (https://memory-alpha.fandom.com/wiki/Abaddon%27s_Repository_of_Lost_Treasures).
It is possible you can go to such a junkyard and either request or trade for an old ship and fix it up and own it. SS Xhora (Cassidy Yates' ship) is described as being very old with no frills and outdated technology.
In "For the Cause," the Xhosa's bridge display graphics are noticeably reminiscent of the screens seen on the bridge of the USS Enterprise on The Original Series, implying that the Xhosa may be a century old. - Background Info
Given the extreme age and outdated technology, it is possible Cassidy acquired her ship via some kind of depot.
Additionally you see Cisco builds his own solar sail ship based on ancient Bajorian designs. There is no ship replicator.
in "Mudd's Women" Mudd and the women travel in:
SPOCK: Sensor reading on the vessel. I make it out as a small class J cargo ship, and his engines are super-heating.
This vessel might be privately owned.
The computer's list of Mudd's offenses:
COMPUTER: Offense record. Smuggling. Sentence suspended. Transport of stolen goods. Purchase of space vessel with counterfeit currency. Sentences, psychiatric treatment, effectiveness disputed.
Mudd certainly did buy a space vessel, but it might have been confiscated when his money was found to be counterfeit.
In "The Trouble With Tribbles" Nils Baris suspects Cyrano Jones could be a Klingon agent:
BARIS: I am going to report fully to the proper authorities that you have given free and complete access to this station to a man who is quite probably a Klingon agent.
KIRK: That's a serious charge. To whom are you referring?
BARIS: To that man who just walked out of here.
KIRK: Cyrano Jones, a Klingon agent?
BARIS: You heard me.
KIRK: I heard you.
SPOCK: He simply could not believe his ears.
KIRK: What evidence do you have against Mister Jones?
BARIS: My assistant here has kept Mister Jones under close surveillance for quite some time, and his actions have been most suspicious. I believe he was involved in that little altercation between your men and the men from
KIRK: Yes, yes. Go on. What else do you have?
DARVIN: Well Captain, I've checked his ship's log, and it seems that he was within the Klingon's sphere of influence less than four months ago.
BARIS: The man is an independent scout, Captain. It is quite possible he is also a Klingon spy.
SPOCK: We have already checked on the background of Mister Cyrano Jones. He is a licensed asteroid locator and prospector. He's never broken the law, at least not severely. For the past seven years, with his one-man spaceship, he's obtained a marginal living by engaging in the buying and selling of rare merchandise, including, unfortunately, tribbles.
This suggest that Jones might own his one-man spaceship.
In "Metamorphosis" Zefram Cochrane went off into space and disappeared in a one-man spaceship that he might have owned.
In the animated episode "the Survivor":
Captain's log, stardate 5143.3. While patrolling the edge of the Romulan Neutral Zone, the Enterprise has changed course to assist a one-man vessel which has fallen victim to a meteor swarm. Sensors indicate the occupant is still alive. [Transporter room]
KIRK: Have you identified the vessel, Mister Spock?
SPOCK: Yes Captain, but you may find it difficult to believe. The vessel is registered to Carter Winston.
MCCOY: That's impossible, Spock. Carter Winston's been missing over five years.
The Enterprise has rescued a living legend, the foremost space trader of our time. Carter Winston has acquired a dozen fortunes only to use his wealth time and again to assist Federation colonies in times of need or disaster.
Thus private ownership of interstellar spaceships happens in the era of TOS and TAS.