How to dye/paint vinyl coated train case?

Not sure painting it would be a good idea as it will no doubt be bumped around a lot (and would scratch off) . How about some fancy travel (or other) decals / stickers to pick up the colors in your other luggage . See the samples below.

Hi, if you plan to use it for traveling, the paint, etc. will not last and just be a mess. Consider having it professionally painted at an auto paint shop. Otherwise leave it and enjoy.

if there are spray paints for vinyl, they’ll be at any home imp. Store.

I painted a multi-colored design on a piece of old luggage for my grandaughter several years ago. I hand-painted a 70s "flower power" design, but you could paint (or stencil) any design you like. Acrylic paint was used and then I applied a couple coats of polyurethane for protection. That was 6 years ago and it still looks great even after being sent on numerous vacations and trips. Good luck!