CMYK, RGB or Pantone?
If there's one piece of advice I can give to people wanting to produce their designs onto accessories, it's to download a chart of colours from the manufacturer's website first. I spent ages choosing colours for a pattern only to have it not turn out as expected once it was digitally printed. It's not the manufacturer's fault, it's just the nature of the beast.
However there are companies out there who produce a handy colour chart so that you can just select a colour with the eye drop picker, within whatever software you are using and you have a digitally safe design. You'll then be able to recolour your artwork, in some software, using the recolour artwork tool and selecting the relevant swatch library, this will go some of the way to converting it. If you need any advice on how to do this, please contact me and I'll be happy to help.