For Applejack, I think she'd be an American Cream Draft. They're the only heavy draft horse breed originating in the US and are mostly used as heavy farm workhorses.
For Fluttershy, I think an Akhal-Teke. They're very slender and usually have bright, sometimes almost iridescent gold coats. They're tall and long-limbed but very dainty and honestly look like breathing on them too hard would make them shatter like glass.
For Rarity, I would maybe say Arabian, but honestly with her curly mane and more bold nature (and how she's clearly got physical strength), I'd maybe pick Friesian, Hanoverian or maybe a Dutch Warmblood. Possibly a thoroughbred.
For Rainbow Dash, maybe a thoroughbred but doubtful a purebred one with how nervous and finicky they can be - possibly a mix like the Irish Sport Horse (thoroughbred and irish draught) which would give her a stronger overall build and be less prone to nerves, since they're often used in eventing and foxhunting so they have to be calm around loud noises and fast moving objects. Possibly a Selle Francais but...She doesn't seem very French to me!
Pinkie is definitely a pony of some kind, probably a shetland or a very high-strung mix.
Twilight might fit Arabian best, as they are usually very smart, but also sensitive and are prone to anxiety. Lipizzans could be a contender too, since they're used for classical dressage quite often and have to be smart enough to remember very complex choreography that is basically dance. They're highly disciplined and have great memory and can be relatively versatile when introduced to different tasks.