What might Carrot and Yamato's Straw Hat color be?
I think for Yamato white is the most likely color for her. Since she has a lot of white from her clothing to her hair and probably her Zoan forms as well.
Considering that Franky is a lighter shade of blue compared to Sanji's I can see something similar with Carrot. Possibly something like a light grass green to represent her Zou and Mink origins. Or a shade of orange to match carrot colors.
What colors might other characters represent if they were to join the crew?
I pretty much have the theory about colors. You see, the 6 main colors of rainbow (red, orange, yellow, green, blue, violet) are the colors of 6 human members of Straw Hats:
Red: Luffy; Orange: Nami; Yellow: Usopp; Green: Zoro; Blue: Sanji; Violet: Robin
Brook’s official color is Black and White, therefore he represents the balancing. However the “Black” represents “stronger”, “White” represents “weaker”. For example:
Red: Luffy
Pink: Chopper
Luffy is stronger than Chopper because pink is lighter shade of red. Also like Franky, his official color is Cyan. Sanji’s official color is blue, although Sanji almost rarely seen in blue colored suits post timeskip. Cyan is lighter shade than Blue, therefore Franky is weaker than Sanji. My theory is further proved when Jinbe’s color is Ochre, and Yellow is Usopp. Ochre is darker shade of Yellow, Jinbe is stronger than Usopp
So, since Yamato is among the strongest ones, her most possible color is Persimmon. It’s darker shade of Orange, and Yamato is obviously stronger than Nami.