In 4 Kids in 5E and 1 Crazy Year, Destiny ships Giovanni/Ashley as well as Maximo/Tiffany.
Jo from Louisa May Alcott's Little Women ships her best friend Laurie (a guy who has a crush on her, but she doesn't reciprocate) with two of her sisters: Meg in Part I, and Beth in Part II. Meg marries Laurie's tutor John Brooke, Beth dies in her youth, and Laurie marries Jo's other sister Amy.
After Miron leaves Dora in book four of Dora Wilk Series, Baal of all (non)people starts shipping them back together, proclaiming himself official candidate for her hand to make Miron jealous. Dora's not too happy about it.
On the other hand, Dora's shipping Olaf and Inga at the same time, with Szelma's help.
Hermione appears to be a Harry/Ginny shipper, to the chagrin of many Harmoniansnote Harry/Hermione shippers.
Some interpreted Ron's "find someone better next time" after Ginny dumps Michael to be a hint that he also supports the pairing, since he was casting Harry an oddly furtive look as he said it. This was on the same page as when Ron also told Harry that he's better off with someone more "cheerful" than Cho. Harry doesn't get the hint, and spends most of the next book afraid Ron will kill him if he makes a move on Ginny.
Harry, meanwhile, ships Ron/Hermione, although he would possibly ship it a bit more if they hadn't chosen the middle of a battle to get together...
In a nod toward shippers in general, there's a moment in Half Blood Prince where Harry and Hermione start joking about the evidence of a relationship between Filch and Madame Pince, the librarian. There's a nod towards this when they're shown standing together later in the book.
Also in Half-Blood Prince, Ginny disapproves of Bill's relationship with Fleur and instead ships Bill/Tonks. She thinks that Mrs. Weasley also ships Bill/Tonks, but actually Mrs. Weasley supports Tonks while she's pining over Remus Lupin. (Ron shipped Bill/Fleur all along, mostly because Fleur mesmerized him.)
Aphrodite, Goddess of Love, is a big Percy/Annabeth shipper in Percy Jackson and the Olympians. Though (understandably) she regards mortals as a soap opera (as Lampshaded by her beau, Ares), and so assures Percy that she won't let things be easy for them, with plenty of Wangst, indecision, etc. In the Sequel SeriesThe Heroes of Olympus, you can add Jason/Piper to the list — and she even goes on about how she thinks couples are cute. "Paris and Helen were a cute couple..."
In The Son of Neptune, Percy ships Hazel and Frank together.
In return, all of the crew of Argo II apparently ship Percy/Annabeth, who act as the steady, established pair the other couples look to as an example. Piper's a standout shipper though as she frequently comments how they make each other happier, better and more complete people. It help's Piper is the daughter of the aforementioned Goddess of Love.
Sam Temple apparently ships LanaXEdilio (Landilio) as stated in the first book:
Sam: You like Lana, don't you?
Edilio: Ur...She's amazing, but—
Sam: Well you can ask her out after we save the world.
In Hunger, both Caine and Diana tease Computer Jack about his budding will-they-won't-they relationship with Brianna.
No one ever shuts up about how Caine loves Diana in Light. Edilio, Sam, Astrid, Virtue and Toto all reference to how if Caine has but one redeemable quality, its that he loves her...To the point Caine had to admit it too. A little bit too late...
In I, Jedi Corran Horn pretty much ships his boss, General Wedge Antilles, with his ex-partner on the Corellian Security Force, Iella Wesseri. In the X-Wing Series, the two had been drawn together gradually, had separated when Iella's vanished husband was found in an Imperial prison, and Wedge had been there for Iella when her husband turned out to be a Manchurian Agent and she had to kill him. But they'd been drawing apart again by this point, and despite Corran telling both of them that they were good for each other, Wedge fell for Qwi Xux while bodyguarding her. Corran was dismayed by this, though to his credit he did know better than to try and separate them.
In Meg Cabot's The Princess Diaries series, Tina Hakim Baba ships Michael/Mia for most of the series, as does Lily in some ways (what with the plots to get them together and all), though the latter is of course influenced by the fact that Michael is her brother.
Actually, Meg seems to do this a lot in her novels: Shari seems so to ship Lizzie and Chaz, Magda and Heather's father ship Heather/Cooper, Luke seems to ship Jenny/Scott (though perhaps leaning closer to matchmaking then strictly shipping), Jen ships Will/Ellie, etc.
Sparhawk very quietly ships Kring/Mirtai in The Elenium trilogy, something he only confides to his wife. The sequel trilogy, the Tamuli, ends with them planning their wedding.
In Safehold Sharleyan ships Hektor and Irys heavily during their trip to Chisholm, and with success. Bonus point for all three being on actual deck of a ship during this.
Pretty much the entire cast of the Chronicles of Prydain ships Taran/Eilonwy, including at least one major villain and Eilonwy's (former) betrothed.
A rather cruel twist on this occurs in Lonely Werewolf Girl, fire-demon Malveria actively (and succesfully) ships human Perky Goth Moonglow with Ordinary College Student Daniel, because Moonglow volunteered to have any hypothetical love with Daniel go unrequited as part of a Deal with the Devil pact earlier in the book and Malveria looks forward to the entertainment it'll bring.
Vex also ships Daniel/Moonglow, to the extent of getting the deal between Moonglow and Malveria annulled.
Malveria does this a lot; one of the subplots of Curse Of The Wolf Girl involved her trying to find a boyfriend for Thrix.
Bilbo — being quite the fan of Aragorn — supports the pairing of him with Arwen; after the quest is over, Frodo notes that the only parts of the tale that pique his interest are the coronation and wedding of Aragorn, and Bilbo expresses his regret at not being able to attend, saying that he had been waiting long for it. Being The Matchmaker of the pair, Galadriel also counts.
During the sequence in the Houses of Healing, Merry seems to be this for Faramir and Éowyn.
In the book, and to a lesser extent the movies, Frodo ships Sam and Rosie. He even invites them to move into his house, since they can't afford one of their own.
The Iliad (a rather dark book to say the least) becomes exponentially funnier if you view Agamemnon as this to Menelaus and Helen. The best part? It's more or less accurate. While other versions of the story give him an ulterior motive, in the original he just wants to help his brother get his wife back, and is willing to raise an entire army and raze her lover's city to the ground to do it.
In the background myth, the entire Greek Army is this. Or at least the leaders: when Helen was to marry, Agamemnon and Odysseus had all the suitors swear that, whoever won her hand, they'd support her marriage with him, no matter what or the cost. Thus the war, fought against the Trojans who are trying to defend the new marriage between Helen and Paris. Yes, this means the greatest human war of the Greek mythology was nothing more than a gigantic example of In-UniverseShip-to-Ship Combat...
In The Belgariad, Garion ships Polgara/Durnik, and possibly Cyradis/Zakath (with restrictions). Ce'Nedra ships Silk/Liselle, to the point of spying on them when Liselle confronts Silk about it. This is one of Polgara's main duties in the prequel, as guardian to the Rivan line (she has to make sure everyone gets together with their destined lover). And the Grolims all ship Polgara/Torak, much to her disgust.
Emma. Most of the plot unfolds because of the heroine's enthusiastic, if completely misguided, shipping.
Most Jane Austen novels have at least a few of these characters, as speculation about who is "attached to" whom is a popular topic for casual conversation. Persuasion even has Ship-to-Ship Combat between Charles and Mary Musgrove, who each want a different Musgrove sister (Louisa and Henrietta respectively) to end up with Frederick Wentworth. Neither one wins. Mrs. Bennet in Pride and Prejudice is another notable one, what with her obsessive desire to see all her daughters married-off as soon as possible.
Both Bob the Skull and Thomas are big HarryxMurphy shippers, even though the two have discussed the reasons they don't want to get into a relationship (Harry doesn't want anything casual, Murphy doesn't want anything serious). Technically, Bob and Thomas are big HarryxAnybody shippers, causing him to distrust their sincerity about Murphy.
Michael Carpenter was also a minor shipper of HarryxSusan by suggesting he marry her before she was infected by the Red Court, though this had more to do with him being a devout Catholic and disapproving of them having sex before marriage.
In Fool Moon, Harry's Subconscious shipped HarryxMurphy when Harry was still involved with Susan.
And by Cold Days Thomas gets just as frustrated at Harry for not immediately seeking out Murphy upon his return to Chicago as he does for Harry not going to see Maggie, his daughter. Adorably, Mouse also shows that he ships HarryxMurphy.
Wicked Lovely: Rianne in the first book tries very hard to fix Keenan and Aislinn up, and in Fragile Eternity Sorcha, of all people, shows some Irial/Niall tendencies.
In Animorphs, Cassie, at a dance, says she hopes Rachel and Tobias will dance with each other. A thousand fans agree. But — uh-oh! The Romantic False Lead, Ax, who everyone ELSE is shipping, causing a thousand fans to groan.
In the first book of the second Circle of Magic quartet, Sandry encourages a romance between her great-uncle Duke Vedris and the famous dancer Yazmín Hebet — an unusual example, with the teenaged protagonist shipping two much older secondary characters. Sandry worries about her long-widowed uncle's health and hopes that love will distract him from worrying too much about the string of murders that she is helping investigate.
In Connie Willis' Passage, Maisie Nellis takes on this role late in the book. After Joanna dies, Maisie tries a couple of times to pair Richard and Kit.
In The Hunger Games, Everyone Can See It when it comes to Peeta and Katniss...everyone being the entire Capitol audience. And while Haymitch encourages Katniss to ratchet up the romance for the sake of effect, it's implied that he ships the two in reality, as well. Unsurprisingly, Haymitch also briefly notices the potential for their Ship Mates Gale and Madge.
Pretty Little Liars: Mike is a big shipper of his sister Aria with his best friend Noel Kahn and is thrilled when they start dating.
The Wereling Trilogy has a rather dark example. The first book, Wounded, has it very clear that Marcy ships Tom with her teenage daughter, Kate, including making Kate dress up fancily while they all have dinner together. Then, it turns out that Marcy kidnapped Tom to secretly turn him into a werewolf and have him mate with Kate, so she would also be a werewolf. When Tom makes it clear that he wants no part in any of that scheme, she sets about trying to messily kill him through the rest of the series. It also turns out that she did pretty much the same shipping scheme with Kate's previous human boyfriend, only killing him after Kate made it clear that she refused to sleep with any werewolf, even him.
In Across the Universe, there are hints that Harley ships Amy and Elder. He seems to enjoy suggesting that Elder hold Amy while they go through the wind tubes, at any rate.
Heroics: Literally everyone in the story ships Cass/AJ in some way.
Hollyleaf does this two times with Lionblaze. The first was with Honeyfern but only to make him be loyal to ThunderClan and get away from Heathertail. The second was with Cinderheart, but she knows he likes her and wants to let them be together.
Rosetail for Bluestar and Thrushpelt...much to Bluestar's discomfort, as she's secretly Oakheart's mate.
Queen Tathilya in The Quest of the Unaligned stuns her son, Prince Alaric, by revealing that he has to get engaged within a few months and suggesting a long list of candidates, with the list headed by the beautiful Gratelle (who apparently makes very good berry tarts). Laeshana, who Alaric actually is in love with, is nowhere on the list.
In the Discworld book Mort, Death appears to ship the eponymous protagonist with his adopted daughter.
Cheri of The Ultra Violets is intent on pairing up her friend Scarlet with Agent Jack Baxter, a secret agent for their arch-nemesis, BeauTek. She considers complications like that to be a minor speed bump on the road of romance.
Cordelia: You two are two of my favorite people. If only you'd get your heads on straight...
First Mate Hiram Miller in Passage to November, despite being a Moment Killer, openly and fervently ships Captain William McTavish with Clara Grace, their freighter's lovely cook.
Miller: And what are you going to do if she leaves?
McTavish: I'll do what I always do.
Miller: And you'll die like the old man, only worse. They'll be hauling you off this boat feet-first and alone.
McTavish: Then that's my business, isn't it?
Miller: If that's the way you want it, fine. Be miserable, and when she's gone, pine away for her for the rest of your life. You know that's just what's going to happen, and it will be all your own damn fault!
It's subtle, but throughout the series Gaius Sextus seems to do everything in his power to put Bernard and Amara and Tavi and Kitai together –- first, having Amara serve as his personal envoy to Bernard as the new Count of Calderon (thereby ensuring that she'll see him regularly), and then when Tavi bullshits his way into having Kitai made an ambassador, makes it his only duty to be her teacher and guide for the following several weeks. The irony here is that Amara thinks she has to hide her marriage to Bernard because of her duty to Gaius and Tavi wants to hide his relationship so Gaius can't exploit it, but he seems to know about and approve of both cases.
Doroga's closing remarks in Furies of Calderon has him say Tavi is "doomed", and he and his wife were just like Kitai and Tavi, which just involved clear disagreeing.
In The Mortal Instruments, Jace and Isabelle both ship their brother Alec with Magnus, despite traditional Shadowhunter prejudices about both homosexuality and Downworlders.
In Vampire Academy, Queen Tatiana and other Royals ship Lissa and Adrian as they're considered more suitable matches for each other than their actual partners Christian (who, despite being a Royal Moroi, is from a disreputable family) and Rose (who is a dhampir).
Razz in Don't Call Me Ishmael! ships Ishmael/Kelly, Scobie/Prue and Miss Tarango/Mr Guthrie. In the third book he briefly ships Ishmael/Cindy. He occasionally tries to play matchmaker as well.
Barkwire, in which the human characters form a "local stray dog" fandom, inevitably contians elements of this.
In A Brother's Price, by the end of the book, Cullen is lobbying his sisters to marry his brother-in-law Jerin's cousin Dail. Apparently, it's being seriously considered by the involved parties.
In The Fault in Our Stars, literally everyone Augustus and Hazel encounter ships them. This includes people they don't even know; a woman dining in the same restaurant offers them a toast, and when they have The Big Damn Kiss, everyone present bursts into applause!
In A Song of Ice and Fire, there is a bit of a variant of this trope where Arya is a firm shipper for her Happily Married parents. When it was suggested that her father Ned was (possibly) in love with Ashara Dayne, Arya actually tries to run away from the Brotherhood.