oooh seriously, if you find a stain that looks like it might be a bit difficult to shift, do nothing until you research!

If its vinyl, do you mean the plastic has sweated? As a plastic, you could try white spirit and then wash in a pure soap like sunlight or lux flakes. Just put something behind the stain (like a folded clean nappy) and then dab the stain from the outside in with the spirit, using another clean cloth to do that. Or you could try drycleaning. *do* check the puffer jacket instructions though, if it says do not dryclean, then don't use white spirit.

Good luck.