Snow White - Hufflepuff
Cinderella - Hufflepuff
Aurora - Hufflepuff? I don't feel like I know her personality enough to sort her
Ariel - Gryffindor
Belle - Gryffindor
Jasmine - Gryffindor
Pocahontas - Ravenclaw
Mulan - Gryffindor
Tiana - Slytherin
Rapunzel - Ravenclaw
Merida - Gryffindor
Anna - Gryffindor
Elsa - Ravenclaw
Moana - Gryffindor

I agree with disnerdtobe that most protagonists are Gryffindors. Even some of the princesses who I didn't choose Gryffindor for definitely have a strong Gryffindor side.

In general I don't think the princesses display Slytherin qualities (cunning and ambition) - the only ones who do are Mulan, Tiana and Merida. Gryffindor qualities outweigh these in Mulan and Merida, but not for Tiana, hence I ended up placing her in Slytherin - she would be a good-type Slytherin though, not a nasty one.

Snow White - Hufflepuff
Cinderella - Hufflepuff
Aurora - Hufflepuff, maybe?
Ariel - Gryffindor
Belle - Gryffindor (she has some of Hufflepuff traits)
Jasmine - Slytherin
Pocahontas - Ravenclaw
Mulan - Gryffindor (she also has some of Hufflepuff traits)
Tiana - Slytherin
Rapunzel - Ravenclaw
Merida - Gryffindor (maybe a bit of Slytherin too)
Anna - Gryffindor
Elsa - Ravenclaw
Moana - Gryffindor

Like what Disnerdtobe and JungleQueen said, most of protagonist are Gryffindor as well. And yeah barely any princesses count as Slytherin. But I count Jasmine and Tiana as Slytherin, as Jasmine is a bit cunning and tend to break rules, and how ambitious Tiana was.

Merida is somewhat a Slytherin but I see her being more dominant in Gryffindor. Belle and Mulan are mix of both Gryffindor and Hufflepuff as they both being typical of brave, fearless heroine but also being so loyal with their beloved ones. We hardly seen any of smartness relate with Elsa but it seems she has good logic and that's why I'm sure that she's Ravenclaw as well.

Snow White- Hufflepuff
Cinderella - Hufflepuff
Aurora -Hufflepuff
Arielle - Ravenclaw( because she is an explorer) or Gryffindor
Belle - same
Jasmine - Gryffindor
Pocahontas - Gryffindor
Mulan - Gryffindor or Slytherin
Tiana - Slytherin
Rapunzel - Ravenclaw
Merida - Gryffindor
Anna - Hufflepuff
Elsa - Ravenclaw

Snow White-Hufflepuff
Cinderella-Hufflepuff (Or Ravenclaw)
Aurora-Hufflepuff
Ariel-Slytherin (Or Gryffindor. Maybe...I don't know...)
Belle-Ravenclaw (Or Hufflepuff)
Jasmine-Slytherin (Or Gryffindor)
Pocahontas-Gryffindor
Mulan-Gryffindor
Tiana-Ravenclaw (Or Gryffindor. This one's actually very hard for me.)
Rapunzel-Slytherin
Merida-Gryffindor
Anna-Slytherin
Elsa-Ravenclaw (Or Gryffindor. Another very hard choice for me.)
Moana-Gryffindor

As you can see, some to most of MY picks are multi-housed. I actually see multiple traits in some of the DP's.