Snow White's dwarfs are probably more than mere sidekicks, the film's full name is Snow White and the Seven Dwarfs after all. They're in the original story, but they're far more important in the film, and the idea of assigning specific personality quirks to each of them is something that defines them and sets it apart from the original material.
Cinderella has a host of animal companions who help to make her life of drudgery under her formidable stepmother and thoughtless stepsisters a little less miserable, but the duo above of Jacques and Gus are two who go the extra mile for Cinderella and will always have a special place in her affections.
Aurora or Sleeping Beauty doesn't really have any clearly defined sidekicks in the way that Snow White and Cinderella did. Given that she doesn't appear that much in the film before being put to sleep that's probably not surprising. Like Snow White and Cinderella she does seem to have an affinity for charming small woodland creatures as shown in the above still from the film.
Ariel would have been lost without her lifelong companion of Flounder and the sage advice of her fussy crustacean chaperone Sebastian. I'm not sure how Flounder handles Ariel being land bound a lot of the time, but Sebastian can operate both in and out of water.
Belle's another Princess who doesn't have a sidekick as such. There's Philippe the horse, but he's as much her father's as he is Belle's. The Beast's transformed animated staff serve the dual purpose of comedy relief and Cuteness Factor, but they're more the Beast's than Belle's, although they did make a brief appearance with Belle in Amy Mebberson's Pocket Princesses strip.
Aladdin is unusual in that the focus isn't on Princess Jasmine, but the street rat Aladdin. He has his own sidekick in the monkey Abu, but that doesn't mean that Jasmine doesn't have one, and that's her protector and playmate the giant tiger Rajah.
In keeping with her at one with nature image Pocahontas was given Flit a hummingbird and Meeko a raccoon as her sidekick/companions in the film. They made an amusing and largely irresistible duo, and often seemed to be starring in their own side story within the film.
Mulan broke new ground gaining a mythological creature as her sidekick. Being voiced by Eddie Murphy made Mushu one of the most amusing sidekicks to date. Although the cricket Cri-Kee was supposed to be Mulan's good luck charm he seemed to become Mushu's very own sidekick as the film played out and the two were inseparable friends.
Tiana doesn't have a sidekick as such. The closest is the Cajun firefly Ray, but he doesn't survive the film (at least not in his accepted form), so I went for the jazz loving, trumpet playing alligator Louis. He is spotted playing with the house band in Tiana's restaurant at the end of The Princess and the Frog and he could also act as the bouncer if things got a little rowdy.
Pascal, Rapunzel's chameleon friend in Tangled was a wonderful sidekick, and rarely ever left the blonde's side or shoulder, which is where he spent a lot of his time. I always wondered just how she met him, because it was clear that his existence was kept a secret from Mother Gothel. I guess being a chameleon he could have climbed up the tower she was kept in. The story of how they first met would make a nice short.
Merida's brothers; Hamish, Hubert and Harris, whether they're in bear or human form are undeniably cute and funny, but they're not sidekicks as such, but they're about the closest I can come in Brave. Another contender is Angus the Clydesdale, but the triplets are much funnier.
And finally we come to Frozen and arguably the most unique sidekick of the lot: Olaf the Snowman, who likes warm hugs and wants to experience summer. Elsa's magic created Olaf, and it's also what gives him his own flurry and means he can survive seasons other than winter, but he's really Anna's sidekick. I guess as long as they give him plenty of warm hugs he can be there for both of them when they need him.