Handwashing for sure. Dollspart sells some different clothes cleaner that are gentle enough to be used on bisque doll clothing and stuff from the 40's and 50's where I specialize in dolls.
Every vinyl/plastic is going to react differently. You know how the acne / removezit cleaners bleach ponies out? Most dolls it works wonderfully on! I unfortunately learned it didn't work on ponies the hard way. Not that there were much in way of warnings, as that was when the product first came out. :Looks forlornly at her Cranberry Muffins, noting that it was AT LEAST her NDS side:
I would consult a Jem forum or such to find out what chemicals are and aren't safe to be sure. Clothing is generally a sure go, I would suggest the dollspart clothing cleaners even for ponywear, just beware of things with red dyes, and clean them separately. Also, do your cleaning like you'd do your own laundry in your sink. Lights with lights, darks with darks, and as said, reds are kept separately. If they bleed, at least they bleed on each other!