Do you have a drill? You'll find it much easier if you pre-drill the hole for your screw to go into. Most materials will fracture or break if you try to force a screw in.
You'll also be better off, if you go an option like chicago screws, to carefully carve out a small indent so that the screw sits flush when you glue it in, that way it'll stop there from being a gap of a millimetre or two between your armour and the medallions.
Another thought is you could use something like split pins
, the head glued into the medallion, and the pin pushed through the vinyl and secured at the back (either sewing over the legs of the pin, gluing, or even gluing a layer of fabric over it.
I'd suggest getting split pins from the hardware store, as your basic stationary ones won't have the strength you want.
Any chance of a reference image, or progress shot of your armour?