Treasure Planet: Just take a look around YouTube or Deviant ART for Jim Hawkins; the poor kid's been paired with nearly every heroine in theatrical animation (and some of the guys); among them, Ariel, Audrey and Kayley, the three most popular crossover girlfriends for him and Dimitri seeming to be the most popular boyfriend, which is not even getting into the OCs. Jim works especially well due to him being one of the few Disney protagonists not to suffer from Artistic Age. He looks and acts like a 14-16 year old, which makes him old enough to ship with late teen and adult characters but also young enough to ship with early teen and preteen characters.
The Once-ler from the 2012 The Lorax adaptation, partly because he's attractive, and partially because there's practically no one in his canon story for him to be shipped with. Because he undergoes a Face–Heel Turn and Evil Costume Switch, the two sides of him are often treated as two entirely separate characters, thus shipping Once-ler and Greed-ler.
Fear from Inside Out has been shipped with every other emotion besides himself in Riley's head, and that's not even counting the OT3 of Fear, Anger, and Disgust. Fear is also paired a lot with female emotion OCs and self-insert characters.
Big Hero 6: Both Hiro AND Tadashi have been shipped with various others on numerous occasions, INCLUDING each other. Some popular characters, excluding the characters from the movie itself:
Tadashi: Elsa, Anna, Nani (Lilo & Stich), Hiccup, and Jack Frost.
Hiro is probably the better example: Violet (The Incredibles), Penny (Bolt), Vanellope, Elsa, Anna, Tip (Home), and Riley (Inside Out). Similarily to Jim from Treasure Planet, he's young enough to be shipped with young characters like Riley but old enough to also be shipped with older teens like Anna from Frozen without causing too many fans be squicked.
Being the protagonist, Tinkerbell from Disney Fairies has been shipped with a lot of the major fairies. She's even paired with her twin sister, Periwinkle.
Flynn and especially Rapunzel from Tangled. Rapunzel was the previous fandom bandwagon before Elsa, and still is shipped to bits. The fact Flynn and Rapunzel are Happily Married is carefully ignored in many fanworks. Some of the more popular Punzie pairings are her and: Elsa, Anna, and Hans from Frozen, the Once-ler from The Lorax, Jack from Rise of the Guardians, and Merida from Brave.
Pocahontas fans loathe shipping Pocohantas with John Rolfe (her real-life love interest, not John Smith like myths say). That doesn't mean people hate shipping John Rolfe himself though. Despite being from one of the less popular Disney sequels, John Rolfe is very popular with shippers and seems to get shipped with every Disney lead.
Marina is a popular character for Non/Disney crossover videos. She's been shipped with a large portion of Disney's traditionally animated characters and a large number of Disney-esque characters as well.
Eris is a ship magnet for villainous characters. She also gets shipped with a lot of heroic characters.
The 90s The Swan Princess animated film often gets overlooked as a mediocre Disney clone however Odette is a very popular character amongst shippers. She gets used a lot in Crossover ShipFan Vids.